OFFSET
0,3
COMMENTS
The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=4) Laguerre triangle L(4; n+m,m) = A062140(n+m,m), n >= 0, is N(4; m,x)/(1-x)^(5+2*m), with the row polynomials N(4; m,x) := Sum_{k=0..m} T(m,k)*x^k.
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(m, k) = [x^k] N(4; m, x), with N(4; m, x) = ((1-x)^(2*m+5))*(d^m/dx^m)((x^m)/(m!*(1-x)^(m+5))).
N(4; m, x) = Sum_{j=0..m} (binomial(m, j)*(2*m+4-j)!/((m+4)!*(m-j)!)*(x^(m-j))*(1-x)^j).
From G. C. Greubel, Mar 03 2025: (Start)
T(n, k) = binomial(n,k)*binomial(n+4,k).
Sum_{k=0..n} (-1)^k*T(n, k) = (1/4)*( (1+(-1)^n)*(-1)^((n+2)/2)*(n^2 + 5*n - 2)*Catalan((n+2)/2)/(n+1) + 8*(1-(-1)^n)*(-1)^((n+1)/2)*Catalan((n+1)/2) ). (End)
EXAMPLE
Triangle begins as:
1;
1, 5;
1, 12, 15;
1, 21, 63, 35;
1, 32, 168, 224, 70;
1, 45, 360, 840, 630, 126;
1, 60, 675, 2400, 3150, 1512, 210;
1, 77, 1155, 5775, 11550, 9702, 3234, 330;
1, 96, 1848, 12320, 34650, 44352, 25872, 6336, 495;
1, 117, 2808, 24024, 90090, 162162, 144144, 61776, 11583, 715;
1, 140, 4095, 43680, 210210, 504504, 630630, 411840, 135135, 20020, 1001;
MATHEMATICA
A062264[n_, k_]:= Binomial[n, k]*Binomial[n+4, k];
Table[A062264[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 03 2025 *)
PROG
(Magma)
A062264:= func< n, k | Binomial(n, k)*Binomial(n+4, k) >;
[A062264(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 03 2025
(SageMath)
def A062264(n, k): return binomial(n, k)*binomial(n+4, k)
print(flatten([[A062264(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 03 2025
CROSSREFS
Family of polynomials (see A062145): A008459 (c=1), A132813 (c=2), A062196 (c=3), A062145 (c=4), this sequence (c=5), A062190 (c=6).
Columns: A028347 (k=2), A104473 (k=3), A104474 (k=4), A104475 (k=5), A027814 (k=6), A103604 (k=7), A104476 (k=8), A104478 (k=9).
Diagonals: A000332 (k=n), A027810 (k=n-1), A105249 (k=n-2), A105250 (k=n-3), A105251 (k=n-4), A105252 (k=n-5), A105253 (k=n-6), A105254 (k=n-7).
Sums: A002694 (row).
KEYWORD
nonn,tabl
AUTHOR
Wolfdieter Lang, Jun 19 2001
STATUS
approved