A084785 - OEIS

A084785

Diagonal of the triangle (A084783) and the self-convolution of the first column (A084784).

1, 2, 5, 16, 66, 348, 2298, 18504, 176841, 1958746, 24661493, 347548376, 5415830272, 92410046544, 1712819553864, 34258146124320, 735267392077962, 16852848083339700, 410809882438699346, 10611174406149372736, 289493459925589039804, 8317946739043065421640

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OFFSET

0,2

COMMENTS

In the triangle (A084783), the diagonal (this sequence) is the self-convolution of the first column (A084784) and the row sums (A084786) gives the differences of the diagonal and the first column.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..350

Chao-Ping Chen, Sharp inequalities and asymptotic series related to Somos' quadratic recurrence constant, Journal of Number Theory, 2016, Volume 172, March 2017, Pages 145-159.

Olivier Golinelli, Remote control system of a binary tree of switches - II. balancing for a perfect binary tree, arXiv:2405.16968 [cs.DM], 2024. See p. 17.

FORMULA

G.f. A(x) satisfies (1+x)^2 = A(x/(1+x))^2/A(x). - Michael Somos, Feb 16 2006

G.f.: A(x) = Product_{n>=1} 1/(1 - n*x)^(1/2^n). - Paul D. Hanna, Jun 16 2010

a(n) ~ (n-1)! / (log(2))^(n+1). - Vaclav Kotesovec, Nov 19 2014

From Peter Bala, May 26 2001: (Start)

O.g.f.: A(x) = exp( Sum_{n >= 1} b(n)*x^n/n ), where b(n) = (-1)^n*Sum_{k = 1..n} k!*Stirling2(n,k)*(-2)^k = A000629(n) = 2*A000670(n) for n >= 1. Cf. A090352.

sqrt(A(x)) = 1/(1 + x)*A(x/(1 + x)) = 1 + x + 2*x^2 + 6*x^3 + 25*x^4 + 137*x^5 + ... is the o.g.f. for A084784. See also A019538. (End)

EXAMPLE

G.f.: A(x) = (1-x)^(-1/2)*(1-2*x)^(-1/4)*(1-3*x)^(-1/8)*(1-4*x)^(-1/16)*... - Paul D. Hanna, Jun 16 2010

MATHEMATICA

nmax = 19; sol = {a[0] -> 1};

Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[(1+x)^2 * A[x] - A[x/(1+x)]^2 + O[x]^(n+1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];

sol /. Rule -> Set;

a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)

With[{m=40}, CoefficientList[Series[Exp[Sum[Sum[(-2)^j*j!*StirlingS2[k, j], {j, k}]*(-x)^k /k, {k, m+1}]], {x, 0, m}], x]] (* G. C. Greubel, Jun 08 2023 *)

PROG

(PARI) A = matrix(25, 25); A[1, 1] = 1; rs = 1; print(1); for (n=2, 25, sc = sum(i=2, n-1, A[i, 1]*A[n+1-i, 1]); A[n, 1] = rs - sc; rs = A[n, 1]; for (k=2, n, A[n, k] = A[n, k-1] + A[n-1, k-1]; rs += A[n, k]); print(A[n, n])); \\ David Wasserman, Jan 06 2005

(PARI) {a(n)=local(A); if(n<0, 0, A=1; for(k=1, n, A=truncate(A+O(x^k))+x*O(x^k); A+=A-(subst(1/A, x, x/(1+x))*(1+x))^-2; ); polcoeff(A, n))} /* Michael Somos, Feb 18 2006 */

(Magma)

m:=40;

f:= func< n, x | Exp((&+[(&+[(-2)^j*Factorial(j)*StirlingSecond(k, j)*(-x)^k/k: j in [1..k]]): k in [1..n+2]])) >;

R<x>:=PowerSeriesRing(Rationals(), m+1); // A084785

Coefficients(R!( f(m, x) )); // G. C. Greubel, Jun 08 2023

(SageMath)

def f(n, x): return exp(sum(sum( (-2)^j*factorial(j)* stirling_number2(k, j)*(-x)^k/k for j in range(1, k+1)) for k in range(1, n+2)))

m=50

def A084785_list(prec):

P.<x> = PowerSeriesRing(QQ, prec)

return P( f(m, x) ).list()

A084785_list(m-9) # G. C. Greubel, Jun 08 2023