A135920 -id:A135920 - cp's OEIS frontend

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.

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

Offset: 1

N. J. A. Sloane

Or, triangle of 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)

			Triangle begins:
  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;
  ...
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}.

L. Carlitz, 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.

Vincenzo Librandi, Rows n = 1..100 of triangle, flattened
Thomas Browning, Counting Parabolic Double Cosets in Symmetric Groups, arXiv:2010.13256 [math.CO], 2020.
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.
José L. Cereceda, Sums of powers of integers and the sequence A304330, arXiv:2405.05268 [math.GM], 2024. See p. 2.
M. W. Coffey and 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)
Qi Fang, Ya-Nan Feng, and Shi-Mei Ma, Alternating runs of permutations and the central factorial numbers, arXiv:2202.13978 [math.CO], 2022.
F. G. Garvan, Higher-order spt functions, Adv. Math. 228 (2011), no. 1, 241-265. - From _N. J. A. Sloane_, Jan 02 2013
Petro Kolosov, Polynomial identities involving central factorial numbers, GitHub, 2024. See p. 6.
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 and J. Novak, Jucys-Murphy Elements and Unitary Matrix Integrals arXiv.0905.1992 [math.CO], 2009-2012.
B. K. Miceli, Two q-Analogues of Poly-Stirling Numbers, J. Integer Seq., 14 (2011), 11.9.6.
John Riordan, Letter, Apr 28 1976.
John Riordan, Letter, Jul 06 1978
Richard P. Stanley, Hook Lengths and Contents.

Columns are A002450, A002451.
Diagonals are A000330 and A060493.
Transpose of A008957.
(0,0)-based version: A269945.
Cf. A008955, A008956, A156289, A135920 (row sums), A204579 (inverse), A000290.

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

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

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 *)
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 *)

PARI
```
T(n,k)=if(1M. F. Hasler, Feb 03 2012
```

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

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

T(n,k) = A156289(n,k)/A001147(k). - Peter Bala, Feb 21 2011
From Peter Bala, Oct 14 2011: (Start)
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.
The n-th row sum = A135920(n). (End)
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. 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.
The production array begins:
  1,  1;
  0,  4,  1;
  0,  0,  9,  1;
  0,  0,  0, 16,  1;
  ... (End)
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
From Kolosov Petro, Jul 26 2023: (Start)
T(n,k) = (1/(2*k)!) * Sum_{j=0..2k} binomial(2k, j)*(-1)^j*(k - j)^(2n).
T(n,k) = (1/(k*(2k-1)!)) * Sum_{j=0..k} (-1)^(k-j)*binomial(2k, k-j)*j^(2n). (End)

More terms from Vladeta Jovovic, Apr 16 2000

A269945 Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + k^2*T(n-1, k), for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 21, 14, 1, 0, 1, 85, 147, 30, 1, 0, 1, 341, 1408, 627, 55, 1, 0, 1, 1365, 13013, 11440, 2002, 91, 1, 0, 1, 5461, 118482, 196053, 61490, 5278, 140, 1, 0, 1, 21845, 1071799, 3255330, 1733303, 251498, 12138, 204, 1

Offset: 0

Peter Luschny, Mar 22 2016

Also known as central factorial numbers T(2*n, 2*k) (cf. A036969).

The analog for the Stirling cycle numbers is A269944.

			Triangle starts:
  [0] [1]
  [1] [0, 1]
  [2] [0, 1,   1]
  [3] [0, 1,   5,    1]
  [4] [0, 1,  21,   14,   1]
  [5] [0, 1,  85,  147,  30,  1]
  [6] [0, 1, 341, 1408, 627, 55, 1]

M. W. Coffey and 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.
Peter Luschny, The P-transform.
Peter Luschny, The Partition Transform -- A SageMath Jupyter Notebook.

Columns k=0..5 give A000007, A000012, A002450(n-1), A002451(n-3), A383838(n-4), A383840(n-5).
Variants are: A008957, A036969.
Cf. A007318 (order 0), A048993 (order 1), A269948 (order 3).
Cf. A000330 (subdiagonal), A002450 (column 2), A135920 (row sums), A269941, A269944 (Stirling cycle), A298851 (central terms).

T := proc(n, k) option remember;
    `if`(n=k, 1,
    `if`(k<0 or k>n, 0,
     T(n-1, k-1) + k^2*T(n-1, k))) end:
for n from 0 to 9 do seq(T(n, k), k=0..n) od;
# Alternatively with the P-transform (cf. A269941):
A269945_row := n -> PTrans(n, n->`if`(n=1, 1, 1/(n*(4*n-2))), (n, k)->(-1)^k*(2*n)!/(2*k)!): seq(print(A269945_row(n)), n=0..8);
# Using the exponential generating function:
egf := 1 + t^2*(cosh(2*sinh(t*x/2)/t));
ser := series(egf, x, 20): cx := n -> coeff(ser, x, 2*n):
Trow := n -> local k; seq((2*n)!*coeff(cx(n), t, 2*(n-k+1)), k = 0..n):
seq(print(Trow(n)), n = 0..9);  # Peter Luschny, Feb 29 2024

T[n_, n_] = 1; T[n_ /; n >= 0, k_] /; 0 <= k < n := T[n, k] = T[n - 1, k - 1] + k^2*T[n - 1, k]; T[, ] = 0; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
(* Jean-François Alcover, Nov 27 2017 *)

# uses[PtransMatrix from A269941]
stirset2 = lambda n: 1 if n == 1 else 1/(n*(4*n-2))
norm = lambda n,k: (-1)^k*factorial(2*n)/factorial(2*k)
M = PtransMatrix(7, stirset2, norm)
for m in M: print(m)

T(n, k) = (-1)^k*((2*n)! / (2*k)!)*P[n, k](s(n)) where P is the P-transform and s(n) = 1/(n*(4*n-2)). The P-transform is defined in the link. Compare also the Sage and Maple implementations below.
T(n, 2) = (4^(n - 1) - 1)/3 for n >= 2 (cf. A002450).
T(n, n-1) = n*(n - 1)*(2*n - 1)/6 for n >= 1 (cf. A000330).
From Fabián Pereyra, Apr 25 2022: (Start)
T(n, k) = (1/(2*k)!)*Sum_{j=0..2*k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n).
T(n, k) = Sum_{j=2*k..2*n} (-k)^(2*n - j)*binomial(2*n, j)*Stirling2(j, 2*k).
T(n, k) = Sum_{j=0..2*n} (-1)^(j - k)*Stirling2(2*n - j, k)*Stirling2(j, k). (End)
T(n, k) = (2*n)! [t^(2*(n-k+1))] [x^(2*n)] (1 + t^2*(cosh(2*sinh(t*x/2)/t))). - Peter Luschny, Feb 29 2024

A135921 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k(k+1)x).

Original entry on oeis.org

1, 1, 3, 13, 81, 669, 6955, 88505, 1346209, 23998521, 493956467, 11596542533, 307301505073, 9110471500693, 299893197116059, 10888674034993905, 433549376981078593, 18833037527449398129, 888439543634687700579

Offset: 0

Paul D. Hanna, Dec 06 2007

nonn

			O.g.f.: A(x) = 1 + x/(1-2x) + x^2/((1-2x)*(1-6x)) + x^3/((1-2x)*(1-6x)*(1-12x)) + x^4/((1-2x)*(1-6x)*(1-12x)*(1-20x)) + ...
Also generated by iterated binomial transforms in the following way:
[1,3,13,81,669,6955,88505,...] = BINOMIAL^2([1,1,5,31,253,2673,34833,..]);
[1,5,31,253,2673,34833,541879,...] = BINOMIAL^4([1,1,7,57,577,7389,...]);
[1,7,57,577,7389,115983,2151493,...] = BINOMIAL^6([1,1,9,91,1101,16497,...]);
[1,9,91,1101,16497,301669,..] = BINOMIAL^8([1,1,11,133,1873,32061,..]);
[1,11,133,1873,32061,666579,...] = BINOMIAL^10([1,1,13,183,2941,56529,...]);
etc.

Cf. A071951, A135920, A124373.

a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-j*(j+1)*x+x*O(x^n))), n)

{a(n) = sum( k=0, n, sum( i=0, k, (-1)^(i+k) * (2*i + 1) * (i*i + i)^n / (k-i)! / (k+i+1)! ))} /* Michael Somos, Feb 25 2012 */

a(n+1) = row sums of A071951. - Michael Somos, Feb 25 2012

G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-(k+1)*(k+2)*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013

A307375 Expansion of Sum_{j>=0} j!x^j / Product_{k=1..j} (1 - k^2x).

Original entry on oeis.org

1, 1, 3, 17, 151, 1893, 31499, 666169, 17351967, 543441005, 20079329875, 861908850561, 42439075349543, 2371469004695797, 149022897087857691, 10448429535366899273, 811758520658841809839, 69463012765807086749949, 6511800419610377560644707, 665560984365147223546851985

Offset: 0

Ilya Gutkovskiy, Apr 06 2019

nonn

a(n) is the number of partitions of [2n] such that the largest element of each block is even. a(3) = 17: 123456, 1234|56, 12356|4, 124|356, 1256|34, 12|3456, 12|34|56, 12|356|4, 13456|2, 134|256, 134|2|56, 1356|24, 1356|2|4, 14|2356, 156|234, 14|2|356, 156|2|34. - Alois P. Heinz, Jun 10 2023

Alois P. Heinz, Table of n, a(n) for n = 0..303

Cf. A000670, A135920, A229234, A363589.

Bisection of A290383 (even part).

b:= proc(n, x, y) option remember; `if`(n=0, 1, `if`(n::odd, 0,
      b(n-1, y, x+1))+b(n-1, y, x)*x+b(n-1, y, x)*y)
    end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..19);  # Alois P. Heinz, Jun 10 2023

nmax = 19; CoefficientList[Series[Sum[j! x^j/Product[(1 - k^2 x), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x]

A369527 Array read by downward antidiagonals: A(n,k) = (k+1)^2*A(n-1,k) + A(n-1,k+1) with A(0,k) = 1, n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 2, 1, 5, 7, 1, 10, 30, 37, 1, 17, 107, 227, 264, 1, 26, 298, 1261, 2169, 2433, 1, 37, 687, 5455, 16804, 25480, 27913, 1, 50, 1382, 18557, 105837, 257073, 358993, 386906, 1, 65, 2515, 52267, 516192, 2209584, 4523241, 5959213, 6346119, 1, 82, 4242, 127477, 2009089, 14913889, 50267233, 90976402, 114813254, 121159373

Offset: 0

Mikhail Kurkov, Jan 25 2024

			Array begins:
====================================================
n\k|    0     1      2       3        4        5 ...
---+------------------------------------------------
0  |    1     1      1       1        1        1 ...
1  |    2     5     10      17       26       37 ...
2  |    7    30    107     298      687     1382 ...
3  |   37   227   1261    5455    18557    52267 ...
4  |  264  2169  16804  105837   516192  2009089 ...
5  | 2433 25480 257073 2209584 14913889 78851808 ...
  ...

Ira M. Gessel, General case of the some R-recursions, answer to question on MathOverflow (2024).

Column k=0 is A135920 (without initial term and with different offset).

A(m, n=m)={my(r=vectorv(m+1), v=vector(n+m+1, k, 1)); r[1] = v[1..n+1];
for(i=1, m, v=vector(#v-1, k, k^2*v[k] + v[k+1]); r[1+i] = v[1..n+1]); Mat(r)}
{ A(5) }

A369595 Array read by downward antidiagonals: A(n,k) = A(n-1,k+2) + Sum_{j=0..k} binomial(k,j)*A(n-1,j) with A(0,k) = 1, n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 2, 1, 3, 7, 1, 5, 14, 37, 1, 9, 30, 89, 264, 1, 17, 68, 227, 737, 2433, 1, 33, 162, 611, 2169, 7696, 27913, 1, 65, 404, 1727, 6695, 25480, 98093, 386906, 1, 129, 1050, 5099, 21573, 87964, 358993, 1490687, 6346119, 1, 257, 2828, 15647, 72287, 315688, 1364681, 5959213, 26542518, 121159373

Offset: 0

Mikhail Kurkov, Jan 27 2024

			Array begins:
=============================================================
n\k|     0     1      2       3       4        5        6 ...
---+---------------------------------------------------------
0  |     1     1      1       1       1        1        1 ...
1  |     2     3      5       9      17       33       65 ...
2  |     7    14     30      68     162      404     1050 ...
3  |    37    89    227     611    1727     5099    15647 ...
4  |   264   737   2169    6695   21573    72287   251109 ...
5  |  2433  7696  25480   87964  315688  1174756  4522480 ...
6  | 27913 98093 358993 1364681 5376121 21901073 92076673 ...
  ...

Cf. A135920.

A(m, n=m)={my(r=vectorv(m+1), v=vector(n+2*m+1, k, 1)); r[1] = v[1..n+1];
for(i=1, m, v=vector(#v-2, k, v[k+2] + sum(j=1, k, binomial(k-1, j-1)*v[j])); r[1+i] = v[1..n+1]); Mat(r)}
{ A(6) }

Conjecture: A(n,0) = A135920(n+1). - Mikhail Kurkov, Oct 27 2024

cp's OEIS Frontend

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.

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A269945 Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + k^2*T(n-1, k), for 0 <= k <= n.

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A135921 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k*(k+1)*x).

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A307375 Expansion of Sum_{j>=0} j!*x^j / Product_{k=1..j} (1 - k^2*x).

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A369527 Array read by downward antidiagonals: A(n,k) = (k+1)^2*A(n-1,k) + A(n-1,k+1) with A(0,k) = 1, n >= 0, k >= 0.

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A369595 Array read by downward antidiagonals: A(n,k) = A(n-1,k+2) + Sum_{j=0..k} binomial(k,j)*A(n-1,j) with A(0,k) = 1, n >= 0, k >= 0.

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A135921 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k(k+1)x).

A307375 Expansion of Sum_{j>=0} j!x^j / Product_{k=1..j} (1 - k^2x).