A174264 Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^(3*n+1)*Sum_{k >= 0} (k*(k+1)*(2*k+1)/6)^n*x^k and p(0, x) = 1, read by rows.

1, 1, 1, 1, 18, 42, 18, 1, 1, 115, 1539, 5065, 5065, 1539, 115, 1, 1, 612, 30369, 359056, 1439038, 2255448, 1439038, 359056, 30369, 612, 1, 1, 3109, 487944, 16069256, 177275075, 808273143, 1688579472, 1688579472, 808273143, 177275075, 16069256, 487944, 3109, 1

Offset

0,5

Links

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

Formula

T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^(3*n+1)*Sum_{k >= 0} (k*(k+1)*(2*k+1)/6)^n*x^k and p(0, x) = 1.

T(n, k) = Sum_{j=0..k+1} (-1)^(k-j+1)*binomial(3*n+1, k-j+1)*(j*(1+j)*(1+2*j)/6)^n, with T(0, k) = T(1, k) = 1. - G. C. Greubel, Mar 25 2022

Example

Irregular triangle begins as:
  1;
  1,   1;
  1,  18,    42,     18,       1;
  1, 115,  1539,   5065,    5065,    1539,     115,      1;
  1, 612, 30369, 359056, 1439038, 2255448, 1439038, 359056, 30369, 612, 1;

Mathematica

(* First program *)
p[n_, x_]:= p[n, x]= If[n==0, 1, (1-x)^(3*n+1)*Sum[(k*(k+1)*(2*k+1)/6)^n*x^k, {k, 0, Infinity}]/x];
Table[CoefficientList[p[n, x], x], {n, 0, 10}]//Flatten
(* Second program *)
T[n_, k_]:= T[n, k]= If[n<2, Binomial[n, k], Sum[(-1)^(k-j+1)*Binomial[3*n+1, k-j +1]*(j*(1+j)*(1+2*j)/6)^n, {j, 0, k+1}]];
Join[{1}, Table[T[n, k], {n, 0, 10}, {k, 0, 3*n-2}]//Flatten] (* G. C. Greubel, Mar 25 2022 *)

Prog

(Sage)
@CachedFunction
def T(n, k):
    if (n<2): return binomial(n, k)
    else: return sum( (-1)^(k-j+1)*binomial(3*n+1, k-j+1)*(j*(1+j)*(1+2*j)/6)^n for j in (0..k+1) )
[1]+flatten([[T(n, k) for k in (0..3*n-2)] for n in (0..10)]) # G. C. Greubel, Mar 25 2022

Crossrefs

Cf. A060187, A154283.

Sequence in context: A285527 A097972 A154284 * A124792 A090605 A318168

Adjacent sequences: A174261 A174262 A174263 * A174265 A174266 A174267

Keyword

nonn,tabf

Author

Roger L. Bagula, Mar 14 2010

Extensions

Edited by G. C. Greubel, Mar 25 2022

Status

approved