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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.
3
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
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
Sequence in context: A285527 A097972 A154284 * A124792 A090605 A318168
KEYWORD
nonn,tabf
AUTHOR
Roger L. Bagula, Mar 14 2010
EXTENSIONS
Edited by G. C. Greubel, Mar 25 2022
STATUS
approved