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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.

%I A036969 #152 May 12 2025 08:28:19
%S A036969 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 A036969 11440,2002,91,1,1,5461,118482,196053,61490,5278,140,1,1,21845,
%U A036969 1071799,3255330,1733303,251498,12138,204,1,1,87381,9668036,53157079,46587905
%N 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.
%C A036969 Or, triangle of central factorial numbers T(2n,2k) (in Riordan's notation).
%C A036969 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 A036969 From _Peter Bala_, Sep 27 2012: (Start)
%C A036969 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 A036969 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)
%D A036969 L. Carlitz, A conjecture concerning Genocchi numbers. Norske Vid. Selsk. Skr. (Trondheim) 1971, no. 9, 4 pp.  [The triangle appears on page 2.]
%D A036969 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D A036969 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.8.
%H A036969 Vincenzo Librandi, <a href="/A036969/b036969.txt"> Rows n = 1..100 of triangle, flattened</a>
%H A036969 Thomas Browning, <a href="https://arxiv.org/abs/2010.13256">Counting Parabolic Double Cosets in Symmetric Groups</a>, arXiv:2010.13256 [math.CO], 2020.
%H A036969 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 A036969 José L. Cereceda, <a href="https://arxiv.org/abs/2405.05268">Sums of powers of integers and the sequence A304330</a>, arXiv:2405.05268 [math.GM], 2024. See p. 2.
%H A036969 M. W. Coffey and 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 A036969 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 A036969 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 A036969 Qi Fang, Ya-Nan Feng, and Shi-Mei Ma, <a href="https://arxiv.org/abs/2202.13978">Alternating runs of permutations and the central factorial numbers</a>, arXiv:2202.13978 [math.CO], 2022.
%H A036969 F. G. Garvan, <a href="http://qseries.org/fgarvan/papers/hspt.pdf">Higher-order spt functions</a>, Adv. Math. 228 (2011), no. 1, 241-265. - From _N. J. A. Sloane_, Jan 02 2013
%H A036969 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/PolynomialIdentitiesInvolvingCentralFactorialNumbers.pdf">Polynomial identities involving central factorial numbers</a>, GitHub, 2024. See p. 6.
%H A036969 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 A036969 S. Matsumoto and 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 A036969 B. K. Miceli, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Miceli/miceli4.html">Two q-Analogues of Poly-Stirling Numbers</a>, J. Integer Seq., 14 (2011), 11.9.6.
%H A036969 John Riordan, <a href="/A002720/a002720_2.pdf">Letter, Apr 28 1976.</a>
%H A036969 John Riordan, <a href="/A001850/a001850_2.pdf">Letter, Jul 06 1978</a>
%H A036969 Richard P. Stanley, <a href="http://www-math.mit.edu/~rstan/transparencies/hooks.pdf">Hook Lengths and Contents</a>.
%F A036969 T(n,k) = A156289(n,k)/A001147(k). - _Peter Bala_, Feb 21 2011
%F A036969 From _Peter Bala_, Oct 14 2011: (Start)
%F A036969 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 + ....
%F A036969 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].
%F A036969 A008955 is a signed version of the inverse.
%F A036969 The n-th row sum = A135920(n). (End)
%F A036969 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 A036969 From _Peter Bala_, Sep 27 2012: (Start)
%F A036969 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 A036969 Put c(n) := (2*n)!/2^n. The column k generating function is (1/c(k))*(E(x)-1)^k = Sum_{n >= k} T(n,k)*x^n/c(n). The inverse array is A204579.
%F A036969 The production array begins:
%F A036969   1,  1;
%F A036969   0,  4,  1;
%F A036969   0,  0,  9,  1;
%F A036969   0,  0,  0, 16,  1;
%F A036969   ... (End)
%F A036969 x^n = Sum_{k=1..n} T(n,k)*Product_{i=0..k-1} (x-i^2), see Stanley link. - _Michel Marcus_, Nov 19 2014; corrected by _Kolosov Petro_, Jul 26 2023
%F A036969 From _Kolosov Petro_, Jul 26 2023: (Start)
%F A036969 T(n,k) = (1/(2*k)!) * Sum_{j=0..2k} binomial(2k, j)*(-1)^j*(k - j)^(2n).
%F A036969 T(n,k) = (1/(k*(2k-1)!)) * Sum_{j=0..k} (-1)^(k-j)*binomial(2k, k-j)*j^(2n). (End)
%e A036969 Triangle begins:
%e A036969   1;
%e A036969   1,    1;
%e A036969   1,    5,      1;
%e A036969   1,   21,     14,      1;
%e A036969   1,   85,    147,     30,     1;
%e A036969   1,  341,   1408,    627,    55,    1;
%e A036969   1, 1365,  13013,  11440,  2002,   91,   1;
%e A036969   1, 5461, 118482, 196053, 61490, 5278, 140, 1;
%e A036969   ...
%e A036969 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 A036969 := proc(n,k) local j; 2*add(j^(2*n)*(-1)^(k-j)/((k-j)!*(k+j)!),j=1..k); end;
%t A036969 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 *)
%t A036969 t1[n_, k_] := (1/(2 k)!) * Sum[Binomial[2 k, j]*(-1)^j*(k - j)^(2 n), {j, 0, 2 k}]; Column[Table[t1[n, k], {n, 1, 10}, {k, 1, n}]] (* _Kolosov Petro_ ,Jul 26 2023 *)
%o A036969 (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 A036969 (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 A036969 (Sage)
%o A036969 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 A036969 for n in (1..7) : print([A036969(n,k) for k in (1..n)]) # Peter Luschny, Feb 03 2012
%o A036969 (Haskell)
%o A036969 a036969 n k = a036969_tabl !! (n-1) (k-1)
%o A036969 a036969_row n = a036969_tabl !! (n-1)
%o A036969 a036969_tabl = iterate f [1] where
%o A036969    f row = zipWith (+)
%o A036969      ([0] ++ row) (zipWith (*) (tail a000290_list) (row ++ [0]))
%o A036969 -- _Reinhard Zumkeller_, Feb 18 2013
%Y A036969 Columns are A002450, A002451.
%Y A036969 Diagonals are A000330 and A060493.
%Y A036969 Transpose of A008957.
%Y A036969 (0,0)-based version: A269945.
%Y A036969 Cf. A008955, A008956, A156289, A135920 (row sums), A204579 (inverse), A000290.
%K A036969 nonn,easy,nice,tabl
%O A036969 1,5
%A A036969 _N. J. A. Sloane_
%E A036969 More terms from _Vladeta Jovovic_, Apr 16 2000