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A036969 Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 < k <= n, T(n,1) = 1. 13
1, 1, 1, 1, 5, 1, 1, 21, 14, 1, 1, 85, 147, 30, 1, 1, 341, 1408, 627, 55, 1, 1, 1365, 13013, 11440, 2002, 91, 1, 1, 5461, 118482, 196053, 61490, 5278, 140, 1, 1, 21845, 1071799, 3255330, 1733303, 251498, 12138, 204, 1, 1, 87381, 9668036, 53157079, 46587905 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Or, triangle central factorial numbers T(2n,2k) (in Riordan's notation).

Can be used to calculate the Bernoulli numbers via the formula B_2n = (1/2)*Sum{k= 1..n, (-1)^(k+1)*(k-1)!*k!*T(n,k)/(2*k+1)}. E.g., n = 1: B_2 = (1/2)*1/3 = 1/6. n = 2: B_4 = (1/2)*(1/3 - 2/5) = -1/30. n = 3: B_6 = (1/2)*(1/3 - 2*5/5 + 2*6/7) = 1/42. - Philippe Deléham, Nov 13 2003

From Peter Bala, Sep 27 2012: (Start)

Generalized Stirling numbers of the second kind. T(n,k) is equal to the number of partitions of the set {1,1',2,2',...,n,n'} into k disjoint nonempty subsets V1,...,Vk such that, for each 1 <= j <= k, if i is the least integer such that either i or i' belongs to Vj then {i,i'} is a subset of Vj. An example is given below.

Thus T(n,k) may be thought of as a two-colored Stirling number of the second kind. See Matsumoto and Novak, who also give another combinatorial interpretation of these numbers.

(End)

REFERENCES

Carlitz, L., A conjecture concerning Genocchi numbers. Norske Vid. Selsk. Skr. (Trondheim) 1971, no. 9, 4 pp.  [The triangle appears on page 2.]

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

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

LINKS

Vincenzo Librandi, Rows n = 1..100 of triangle, flattened

P. L. Butzer, M. Schmidt, E. L. Stark and L. Vogt. Central factorial numbers; their main properties and some applications, Num. Funct. Anal. Optim., 10 (1989) 419-488.

M. W. Coffey, 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.

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

D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy)

F. G. Garvan, Higher-order spt functions, Adv. Math. 228 (2011), no. 1, 241-265. - From N. J. A. Sloane, Jan 02 2013

P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

John Riordan, Letter, Apr 28 1976.

J. Riordan, Letter, Jul 06 1978

S. Matsumoto, J. Novak, Jucys-Murphy Elements and Unitary Matrix Integrals arXiv.0905.1992 [math.CO], 2009-2012.

Richard P. Stanley, Hook Lengths and Contents.

FORMULA

T(n,k) = A156289(n,k)/A001147(k). - Peter Bala, Feb 21 2011

O.g.f.: sum {n>=1} x^n*t^n/Product {k = 1..n} (1-k^2*t^2) = x*t + (x+x^2)*t^2 + (x+5*x^2+x^3)*t^3 + .... Define polynomials x^[2*n] = product {k = 0..n-1} (x^2-k^2). This triangle gives the coefficients in the expansion of the monomials x^(2*n) as a linear combination of x^[2*m], 1 <= m <= n. For example, row 4 gives x^8 = x^[2] + 21*x^[4] + 14*x^[6] + x^[8]. A008955 is a signed version of the inverse. n-th row sum = A135920(n). - Peter Bala, Oct 14 2011

T(n,k) = (2/(2*k)!)*Sum_{j=0..k-1} (-1)^(j+k+1) * binomial(2*k,j+k+1) * (j+1)^(2*n). This formula is valid for n >= 0 and 0 <= k <= n. - Peter Luschny, Feb 03 2012

From Peter Bala, Sep 27 2012: (Start)

Let E(x) = cosh(sqrt(2*x)) = sum {n >= 0} x^n/{(2*n)!/2^n}. A generating function for the triangle is E(t*(E(x)-1)) = 1 + t*x + t*(1+t)*x^2/6 + t*(1+5*t+t^2)*x^3/90 + ..., where the sequence of denominators [1,1,6,90,...] is given by (2*n)!/2^n. Cf. A008277 which has generating function exp(t*(exp(x)-1)). An e.g.f. is E(t*(E(x^2/2)-1)) = 1 + t*x^2/2! + t*(1+t)*x^4/4! + t*(1+5*t+t^2)*x^6/6! + ....

Put c(n) := (2*n)!/2^n. Column k generating function is 1/c(k)*(E(x)-1)^k = sum {n = k..inf} T(n,k)*x^n/c(n). Inverse array is A204579.

Production array begins

1...1

0...4...1

0...0...9...1

0...0...0..16...1

...

(End)

x^n = T(n,k)*Product_{i=0..k} (x-i^2), see Stanley link. - Michel Marcus, Nov 19 2014

EXAMPLE

Triangle begins:

  1

  1   1

  1   5   1

  1  21  14   1

  1  85 147  30   1

  ...

T(3,2) = 5: The five set partitions into two sets are {1,1',2,2'}{3,3'}, {1,1',3,3'}{2,2'}, {1,1'}{2,2',3,3'}, {1,1',3}{2,2',3'} and {1,1',3'}{2,2',3}.

MAPLE

A036969 := proc(n, k) local j; 2*add(j^(2*n)*(-1)^(k-j)/((k-j)!*(k+j)!), j=1..k); end;

MATHEMATICA

t[n_, k_] := 2*Sum[j^(2*n)*(-1)^(k-j)/((k-j)!*(k+j)!), {j, 1, k}]; Flatten[ Table[t[n, k], {n, 1, 10}, {k, 1, n}]] (* Jean-François Alcover, Oct 11 2011 *)

PROG

(PARI) T(n, k)=if(1<k && k<=n, T(n-1, k-1) + k^2*T(n-1, k), k==1) \\ for illustrative purpose, not efficient ; M. F. Hasler, Feb 03 2012

(PARI) T(n, k)=2*sum(j=1, k, (-1)^(k-j)*j^(2*n)/(k-j)!/(k+j)!)  \\ M. F. Hasler, Feb 03 2012

(Sage)

def A036969(n, k) : return (2/factorial(2*k))*add((-1)^j*binomial(2*k, j)*(k-j)^(2*n) for j in (0..k))

for n in (1..7) : print([A036969(n, k) for k in (1..n)]) # Peter Luschny, Feb 03 2012

(Haskell)

a036969 n k = a036969_tabl !! (n-1) (k-1)

a036969_row n = a036969_tabl !! (n-1)

a036969_tabl = iterate f [1] where

   f row = zipWith (+)

     ([0] ++ row) (zipWith (*) (tail a000290_list) (row ++ [0]))

-- Reinhard Zumkeller, Feb 18 2013

CROSSREFS

Diagonals are A002450, A002451, A000330 and A060493.

Transpose of A008957. Cf. A008955, A008956, A156289, A135920 (row sums), A204579 (inverse), A000290.

Sequence in context: A171243 A111577 A176242 * A080249 A333143 A157154

Adjacent sequences:  A036966 A036967 A036968 * A036970 A036971 A036972

KEYWORD

nonn,easy,nice,tabl

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, Apr 16 2000

STATUS

approved

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Last modified August 4 20:00 EDT 2020. Contains 336202 sequences. (Running on oeis4.)