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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

%I

%S 1,1,1,1,5,1,1,21,14,1,1,85,147,30,1,1,341,1408,627,55,1,1,1365,13013,

%T 11440,2002,91,1,1,5461,118482,196053,61490,5278,140,1,1,21845,

%U 1071799,3255330,1733303,251498,12138,204,1,1,87381,9668036,53157079,46587905

%N 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.

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

%C 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

%C From _Peter Bala_, Sep 27 2012: (Start)

%C 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.

%C 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.

%C (End)

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

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

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

%H Vincenzo Librandi, <a href="/A036969/b036969.txt"> Rows n = 1..100 of triangle, flattened</a>

%H P. L. Butzer, M. Schmidt, E. L. Stark and L. Vogt. <a href="http://dx.doi.org/10.1080/01630568908816313">Central factorial numbers; their main properties and some applications</a>, Num. Funct. Anal. Optim., 10 (1989) 419-488.

%H M. W. Coffey, M. C. Lettington, <a href="http://arxiv.org/abs/1510.05402">On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat</a>, arXiv:1510.05402 [math.NT], 2015.

%H D. Dumont, <a href="http://dx.doi.org/10.1215/S0012-7094-74-04134-9">Interprétations combinatoires des nombres de Genocchi</a>, Duke Math. J., 41 (1974), 305-318.

%H D. Dumont, <a href="/A001469/a001469_3.pdf">Interprétations combinatoires des nombres de Genocchi</a>, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy)

%H F. G. Garvan, <a href="http://qseries.org/fgarvan/papers/hspt.pdf">Higher-order spt functions, Adv. Math. 228 (2011), no. 1, 241-265. - From _N. J. A. Sloane_, Jan 02 2013

%H P. A. MacMahon, <a href="https://doi.org/10.1112/plms/s2-19.1.75">Divisors of numbers and their continuations in the theory of partitions</a>, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

%H John Riordan, <a href="/A002720/a002720_2.pdf">Letter, Apr 28 1976.</a>

%H J. Riordan, <a href="/A001850/a001850_2.pdf">Letter, Jul 06 1978</a>

%H S. Matsumoto, J. Novak, <a href="http://arxiv.org/abs/0905.1992">Jucys-Murphy Elements and Unitary Matrix Integrals</a> arXiv.0905.1992 [math.CO], 2009-2012.

%H Richard P. Stanley, <a href="http://www-math.mit.edu/~rstan/transparencies/hooks.pdf">Hook Lengths and Contents</a>.

%F T(n,k) = A156289(n,k)/A001147(k). - _Peter Bala_, Feb 21 2011

%F 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

%F 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

%F From _Peter Bala_, Sep 27 2012: (Start)

%F 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! + ....

%F 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.

%F Production array begins

%F 1...1

%F 0...4...1

%F 0...0...9...1

%F 0...0...0..16...1

%F ...

%F (End)

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

%e Triangle begins:

%e 1

%e 1 1

%e 1 5 1

%e 1 21 14 1

%e 1 85 147 30 1

%e ...

%e 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}.

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

%t 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 *)

%o (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

%o (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

%o (Sage)

%o 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))

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

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

%o a036969_row n = a036969_tabl !! (n-1)

%o a036969_tabl = iterate f [1] where

%o f row = zipWith (+)

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

%o -- _Reinhard Zumkeller_, Feb 18 2013

%Y Diagonals are A002450, A002451, A000330 and A060493.

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

%K nonn,easy,nice,tabl

%O 1,5

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, Apr 16 2000

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Last modified September 20 10:24 EDT 2020. Contains 337264 sequences. (Running on oeis4.)