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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. 12
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

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

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.

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

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 A157154 A022168

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 18 14:09 EDT 2017. Contains 290720 sequences.