A204579 Triangle read by rows: matrix inverse of the central factorial numbers T(2*n, 2*k) (A036969).

1, -1, 1, 4, -5, 1, -36, 49, -14, 1, 576, -820, 273, -30, 1, -14400, 21076, -7645, 1023, -55, 1, 518400, -773136, 296296, -44473, 3003, -91, 1, -25401600, 38402064, -15291640, 2475473, -191620, 7462, -140, 1, 1625702400, -2483133696, 1017067024, -173721912, 14739153, -669188, 16422, -204, 1

Offset: 1

Author

M. F. Hasler, Feb 03 2012

Keywords

• sign
• tabl

Comments

This is a signed version of A008955 with rows in reverse order. - Peter Luschny, Feb 04 2012

Examples

			Triangle starts:
  [1]         1;
  [2]        -1,        1;
  [3]         4,       -5,         1;
  [4]       -36,       49,       -14,       1;
  [5]       576,     -820,       273,     -30,       1;
  [6]    -14400,    21076,     -7645,    1023,     -55,    1;
  [7]    518400,  -773136,    296296,  -44473,    3003,  -91,    1;
  [8] -25401600, 38402064, -15291640, 2475473, -191620, 7462, -140, 1;
		

Links

• José L. Cereceda, Sums of powers of integers and the sequence A304330, arXiv:2405.05268 [math.GM], 2024. See p. 6.
• Mark W. Coffey and Matthew 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.

Crossrefs

Cf. A036969, A008955, A008275, A121408, A001044 (column 1), A101686 (alternating row sums), A234324 (central terms).

Programs

Maple

# From Peter Luschny, Feb 29 2024: (Start)
ogf := n -> local j; z^2*mul(z^2 - j^2, j = 1..n-1):
Trow := n -> local k; seq(coeff(expand(ogf(n)), z, 2*k), k = 1..n):
# Alternative:
f := w -> (w^sqrt(t) + w^(-sqrt(t)))/2: egf := f((x/2 + sqrt(1 + (x/2)^2))^2):
ser := series(egf, x, 20): cx := n -> coeff(ser, x, 2*n):
Trow := n -> local k; seq((2*n)!*coeff(cx(n), t, k), k = 1..n):  # (End)
# Assuming offset 0:
rowpoly := n -> (-1)^n * pochhammer(1 - sqrt(x), n) * pochhammer(1 + sqrt(x), n):
row := n -> local k; seq(coeff(expand(rowpoly(n)), x, k), k = 0..n):
seq(print(row(n)), n = 0..7);  # Peter Luschny, Aug 03 2024

Mathematica

rows = 10;
t[n_, k_] := 2*Sum[j^(2*n)*(-1)^(k - j)/((k - j)!*(k + j)!), {j, 1, k}];
T = Table[t[n, k], {n, 1, rows}, {k, 1, rows}] // Inverse;
Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 14 2018 *)

PARI

select(concat(Vec(matrix(10,10,n,k,T(n,k)/*from A036969*/)~^-1)), x->x)

Sage

def A204579(n, k): return (-1)^(n-k)*A008955(n, n-k)
for n in (0..7): print([A204579(n, k) for k in (0..n)]) # Peter Luschny, Feb 05 2012

Formula

T(n, k) = (-1)^(n-k)*A008955(n, n-k). - Peter Luschny, Feb 05 2012

T(n, k) = Sum_{i=k-n..n-k} (-1)^(n-k+i)*s(n,k+i)*s(n,k-i) = Sum_{i=0..2*k} (-1)^(n+i)*s(n,i)*s(n,2*k-i), where s(n,k) are Stirling numbers of the first kind, A048994. - Mircea Merca, Apr 07 2012

From Peter Bala, Aug 29 2012: (Start)

T(n, k) = T(n-1, k-1) - (n-1)^2*T(n-1, k). (Recurrence equation.)

Let E(x) = cosh(sqrt(2*x)) = Sum_{n >= 0} x^n/{(2*n)!/2^n} and

L(x) = 2*{arcsinh(sqrt(x/2))}^2 = Sum_{n >=1} (-1)^n*(n-1)!^2*x^n/{(2*n)!/2^n}.

L(x) is the compositional inverse of E(x) - 1.

A generating function for the triangle is E(t*L(x)) = 1 + t*x + t*(-1 + t)*x^2/6 + t*(4 - 5*t + t^2)*x^3/90 + ..., where the sequence of denominators [1,1,6,90,...] is given by (2*n)!/2^n. Cf. A008275 with generating function exp(t*log(1+x)).

The e.g.f. is E(t*L(x^2/2)) = cosh(2*sqrt(t)*arcsinh(x/2)) = 1 + t*x^2/2! + t*(t-1)*x^4/4! + t*(t-1)*(t-4)*x^6/6! + .... (End)

From Peter Luschny, Feb 29 2024: (Start)

T(n, k) = [z^(2*k)] z^2*Product_{j=1..n-1} (z^2 - j^2).

T(n, k) = (2*n)! * [t^k] [x^(2*n)] (w^sqrt(t) + w^(-sqrt(t)))/2 where w = (x/2 + sqrt(1 + (x/2)^2))^2. (End)

T(n, k) = [x^k] (-1)^n * Pochhammer(1 - sqrt(x), n) * Pochhammer(1 + sqrt(x), n), assuming offset 0. - Peter Luschny, Aug 03 2024

Integral_{0..oo} x^s / (cosh(x))^(2*n) dx = (2^(2*n - s - 1) * s! * (-1)^(n-1)) / (2*n - 1)!)*Sum_{k=1..n} T(n,k)*DirichletEta(s - 2*k + 2). - Ammar Khatab, Apr 11 2025

Extensions

Typo in data corrected by Peter Luschny, Feb 05 2012