A008957 Triangle of central factorial numbers T(2*n,2*n-2*k), k >= 0, n >= 1 (in Riordan's notation).

1, 1, 1, 1, 5, 1, 1, 14, 21, 1, 1, 30, 147, 85, 1, 1, 55, 627, 1408, 341, 1, 1, 91, 2002, 11440, 13013, 1365, 1, 1, 140, 5278, 61490, 196053, 118482, 5461, 1, 1, 204, 12138, 251498, 1733303, 3255330, 1071799, 21845, 1, 1, 285, 25194, 846260, 10787231

Offset

1,5

Comments

D. E. Knuth [1992] on page 10 gives the formula Sum n^(2m-1) = Sum_{k=1..m} (2k-1)! T(2m,2k) binomial(n+k, 2k) where T(m, k) is the central factorial number of the second kind. - Michael Somos, May 08 2018

References

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217, Table 6.2(a).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.8.

Links

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

Feryal Alayont and Nicholas Krzywonos, Rook Polynomials in Three and Higher Dimensions, 2012. [From N. J. A. Sloane, Jan 02 2013]

Dominique Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J. 41(2) (1974), 305-318.

Andrii Husiev, Extended Central Factorial Numbers and the Flickering Operator, arXiv:2605.06689 [math.GM], 2026. See pp. 1-2, 7-8.

Donald E. Knuth, Johann Faulhaber and Sums of Powers, arXiv:math/9207222, Jul 1992. See bottom of page 10. [From Michael Somos, May 08 2018]

Petro Kolosov, Polynomial identities involving central factorial numbers, GitHub, 2024. See p. 6.

Petro Kolosov, Sums of powers via central finite differences and Newton's formula (2026). See p. 4.

Formula

From Michael Somos, May 08 2018: (Start)

T(n, k) = T(n-1, k-1) + k^2 * T(n-1, k), where T(n, n) = T(n, 1) = 1.

E.g.f.: x^2 * (cosh(sinh(y*x/2) / (x/2)) - 1) = (1*x^2)*y^2/2! + (1*x^2 + 1*x^4)*y^4/4! + (1*x^2 + 5*x^4 + 1*x^6)*y^6/6! + (1*x^2 + 14*x^4 + 21*x^6 + 1*x^8)*y^8/8! + ... (End)

From Husiev Andrii Alekseevich, Apr 01 2026: (Start)

T(n, k) = (1/(2n-2k+1)!) * Sum_{i=0..2n-2k+1} (-1)^(2n-2k+1-i) * binomial(2n-2k+1, i) * (i-n+k)^(2n-1).

T(n, k) = Sum_{j=0..2n-1} binomial(2n-1, j) * (k-n)^(2n-1-j) * Stirling2(j, 2n-2k+1).

(End)

Example

The triangle starts:
  1;
  1,  1;
  1,  5,    1;
  1, 14,   21,     1;
  1, 30,  147,    85,     1;
  1, 55,  627,  1408,   341,    1;
  1, 91, 2002, 11440, 13013, 1365, 1;

Maple

A036969 := proc(n, k) local j; 2*add(j^(2*n)*(-1)^(k-j)/((k-j)!*(k+j)!), j=1..k); end; # Gives rows of triangle in reversed order

Mathematica

t[n_, n_] = t[n_, 1] = 1;
t[n_, k_] := t[n-1, k-1] + k^2 t[n-1, k];
Flatten[Table[t[n, k], {n, 1, 10}, {k, n, 1, -1}]][[1 ;; 50]] (* Jean-François Alcover, Jun 16 2011 *)

Prog

(Haskell)
a008957 n k = a008957_tabl !! (n-1) (k-1)
a008957_row n = a008957_tabl !! (n-1)
a008957_tabl = map reverse a036969_tabl
-- Reinhard Zumkeller, Feb 18 2013
(PARI) {T(n, k) = if( n<1 || k>n, 0, n==k || k==1, 1, T(n-1, k-1) + k^2 * T(n-1, k))}; \\ Michael Somos, May 08 2018
(SageMath)
def A008957(n, k):
    m = n - k
    return 2*sum((-1)^(j+m)*(j+1)^(2*n)/(factorial(j+m+2)*factorial(m-j)) for j in (0..m))
for n in (1..7): print([A008957(n, k) for k in (1..n)]) # Peter Luschny, May 10 2018

Crossrefs

Row reversed version of A036969. (0,0)-based version: A269945.

Cf. A008955.

Cf. A394582 (Parent array, of which this triangle is the odd-column component).

Cf. A394813 (The even-column counterpart; together they form A394582).

Sequence in context: A380179 A144438 A157207 * A389263 A136267 A109960

Adjacent sequences: A008954 A008955 A008956 * A008958 A008959 A008960

Keyword

nonn,nice,easy,tabl

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Apr 16 2000

Status

approved