OFFSET
0,9
COMMENTS
LINKS
M. W. Coffey and M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.
Peter Luschny, The P-transform.
Peter Luschny, The Partition Transform -- A SageMath Jupyter Notebook.
FORMULA
T(n, k) = (-1)^k*((2*n)! / (2*k)!)*P[n, k](s(n)) where P is the P-transform and s(n) = 1/(n*(4*n-2)). The P-transform is defined in the link. Compare also the Sage and Maple implementations below.
T(n, 2) = (4^(n - 1) - 1)/3 for n >= 2 (cf. A002450).
T(n, n-1) = n*(n - 1)*(2*n - 1)/6 for n >= 1 (cf. A000330).
From Fabián Pereyra, Apr 25 2022: (Start)
T(n, k) = (1/(2*k)!)*Sum_{j=0..2*k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n).
T(n, k) = Sum_{j=2*k..2*n} (-k)^(2*n - j)*binomial(2*n, j)*Stirling2(j, 2*k).
T(n, k) = Sum_{j=0..2*n} (-1)^(j - k)*Stirling2(2*n - j, k)*Stirling2(j, k). (End)
T(n, k) = (2*n)! [t^(2*(n-k+1))] [x^(2*n)] (1 + t^2*(cosh(2*sinh(t*x/2)/t))). - Peter Luschny, Feb 29 2024
EXAMPLE
Triangle starts:
[0] [1]
[1] [0, 1]
[2] [0, 1, 1]
[3] [0, 1, 5, 1]
[4] [0, 1, 21, 14, 1]
[5] [0, 1, 85, 147, 30, 1]
[6] [0, 1, 341, 1408, 627, 55, 1]
MAPLE
T := proc(n, k) option remember;
`if`(n=k, 1,
`if`(k<0 or k>n, 0,
T(n-1, k-1) + k^2*T(n-1, k))) end:
for n from 0 to 9 do seq(T(n, k), k=0..n) od;
# Alternatively with the P-transform (cf. A269941):
A269945_row := n -> PTrans(n, n->`if`(n=1, 1, 1/(n*(4*n-2))), (n, k)->(-1)^k*(2*n)!/(2*k)!): seq(print(A269945_row(n)), n=0..8);
# Using the exponential generating function:
egf := 1 + t^2*(cosh(2*sinh(t*x/2)/t));
ser := series(egf, x, 20): cx := n -> coeff(ser, x, 2*n):
Trow := n -> local k; seq((2*n)!*coeff(cx(n), t, 2*(n-k+1)), k = 0..n):
seq(print(Trow(n)), n = 0..9); # Peter Luschny, Feb 29 2024
MATHEMATICA
T[n_, n_] = 1; T[n_ /; n >= 0, k_] /; 0 <= k < n := T[n, k] = T[n - 1, k - 1] + k^2*T[n - 1, k]; T[_, _] = 0; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
(* Jean-François Alcover, Nov 27 2017 *)
PROG
(Sage) # uses[PtransMatrix from A269941]
stirset2 = lambda n: 1 if n == 1 else 1/(n*(4*n-2))
norm = lambda n, k: (-1)^k*factorial(2*n)/factorial(2*k)
M = PtransMatrix(7, stirset2, norm)
for m in M: print(m)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Mar 22 2016
STATUS
approved