A062264 Coefficient triangle of certain polynomials N(4; m,x).

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

Offset: 0

Author

Wolfdieter Lang, Jun 19 2001

Keywords

• nonn
• tabl

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.

Examples

			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;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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).

Programs

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

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 *)

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

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)