A002453 -id:A002453 - cp's OEIS frontend

A136630 Triangular array: T(n,k) counts the partitions of the set [n] into k odd sized blocks.

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 4, 0, 1, 0, 1, 0, 10, 0, 1, 0, 0, 16, 0, 20, 0, 1, 0, 1, 0, 91, 0, 35, 0, 1, 0, 0, 64, 0, 336, 0, 56, 0, 1, 0, 1, 0, 820, 0, 966, 0, 84, 0, 1, 0, 0, 256, 0, 5440, 0, 2352, 0, 120, 0, 1, 0, 1, 0, 7381, 0, 24970, 0, 5082, 0, 165, 0, 1, 0, 0, 1024, 0, 87296, 0

Offset: 0

Paul D. Hanna, Jan 14 2008

For partitions into blocks of even size see A156289.
Essentially the unsigned matrix inverse of triangle A121408.
From Peter Bala, Jul 28 2014: (Start)
Define a polynomial sequence x_(n) by setting x_(0) = 1 and for n = 1,2,... setting x_(n) = x*(x + n - 2)*(x + n - 4)*...*(x + n - 2*(n - 1)). Then this table is the triangle of connection constants for expressing the monomial polynomials x^n in terms of the basis x_(k), that is, x^n = sum {k = 0..n} T(n,k)*x_(k) for n = 0,1,2,.... An example is given below.
Let M denote the lower unit triangular array A119467 and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/ having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle, omitting the first row and column, equals the infinite matrix product M(0)*M(1)*M(2)*.... (End)
Also the Bell transform of A000035(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			Triangle begins:
  1;
  0, 1;
  0, 0,   1;
  0, 1,   0,    1;
  0, 0,   4,    0,    1;
  0, 1,   0,   10,    0,     1;
  0, 0,  16,    0,   20,     0,    1;
  0, 1,   0,   91,    0,    35,    0,    1;
  0, 0,  64,    0,  336,     0,   56,    0,   1;
  0, 1,   0,  820,    0,   966,    0,   84,   0,   1;
  0, 0, 256,    0, 5440,     0, 2352,    0, 120,   0, 1;
  0, 1,   0, 7381,    0, 24970,    0, 5082,   0, 165, 0, 1;
T(5,3) = 10. The ten partitions of the set [5] into 3 odd-sized blocks are
(1)(2)(345), (1)(3)(245), (1)(4)(235), (1)(5)(234), (2)(3)(145),
(2)(4)(135), (2)(5)(134), (3)(4)(125), (3)(5)(124), (4)(5)(123).
Connection constants: Row 5 = [0,1,0,10,0,1]. Hence, with the polynomial sequence x_(n) as defined in the Comments section we have x^5 = x_(1) + 10*x_(3) + x_(5) = x + 10*x*(x+1)*(x-1) + x*(x+3)*(x+1)*(x-1)*(x-3).

L. Comtet, Analyse Combinatoire, Presses Univ. de France, 1970, Vol. II, pages 61-62.
L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 225-226.

Ch. A. Charalambides, Central factorial numbers and related expansions, Fib. Quarterly, Vol. 19, No 5, Dec 1981, pp. 451-456.
Feng Qi and Peter Taylor, Series expansions for powers of sinc function and closed-form expressions for specific partial Bell polynomials, Appl. Anal. Disc. Math. (2024) Vol. 18, No. 1, 1-24. See p. 13.

Cf. A121408; A136631 (antidiagonal sums), A003724 (row sums), A136632; A002452 (column 3), A002453 (column 5); A008958 (central factorial triangle), A156289. A185690, A196776.

A136630 := proc (n, k) option remember; if k < 0 or n < k then 0 elif k = n then 1 else procname(n-2, k-2) + k^2*procname(n-2, k) end if end proc: seq(seq(A136630(n, k), k = 1 .. n), n = 1 .. 12); # Peter Bala, Jul 27 2014
# The function BellMatrix is defined in A264428.
BellMatrix(n -> (n+1) mod 2, 9); # Peter Luschny, Jan 27 2016

t[n_, k_] := Coefficient[ x^k/Product[ 1 - (2*j + k - 2*Quotient[k, 2])^2*x^2, {j, 0, k/2}] + x*O[x]^n, x, n]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 22 2013, after Pari *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 13;
M = BellMatrix[Mod[#+1, 2]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

{T(n,k)=polcoeff(x^k/prod(j=0,k\2,1-(2*j+k-2*(k\2))^2*x^2 +x*O(x^n)),n)}

G.f. for column k: x^k/Product_{j=0..floor(k/2)} (1 - (2*j + k-2*floor(k/2))^2 * x^2).
G.f. for column 2*k: x^(2*k)/Product_{j=0..k} (1 - (2*j)^2*x^2).
G.f. for column 2*k+1: x^(2*k+1)/Product_{j=0..k} (1 - (2*j+1)^2*x^2).
From Peter Bala, Feb 21 2011 (Start)
T(n,k) = 1/(2^k*k!)*Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*(2*j-k)^n,
Recurrence relation T(n+2,k) = T(n,k-2) + k^2*T(n,k).
E.g.f.: F(x,z) = exp(x*sinh(z)) = Sum_{n>=0} R(n,x)*z^n/n! = 1 + x*z + x^2*z^2/2! + (x+x^3)*z^3/3! + ....
The row polynomials R(n,x) begin
R(1,x) = x
R(2,x) = x^2
R(3,x) = x+x^3.
The e.g.f. F(x,z) satisfies the partial differential equation d^2/dz^2(F) = x^2*F + x*F' + x^2*F'' where ' denotes differentiation w.r.t. x.
Hence the row polynomials satisfy the recurrence relation R(n+2,x) = x^2*R(n,x) + x*R'(n,x) + x^2*R''(n,x) with R(0,x) = 1.
The recurrence relation for T(n,k) given above follows from this.
(End)
For the corresponding triangle of ordered partitions into odd-sized blocks see A196776. Let P denote Pascal's triangle A070318 and put M = 1/2*(P-P^-1). M is A162590 (see also A131047). Then the first column of exp(t*M) lists the row polynomials for the present triangle. - Peter Bala, Oct 06 2011
Row generating polynomials equal D^n(exp(x*t)) evaluated at x = 0, where D is the operator sqrt(1+x^2)*d/dx. Cf. A196776. - Peter Bala, Dec 06 2011
From Peter Bala, Jul 28 2014: (Start)
E.g.f.: exp(t*sinh(x)) = 1 + t*x + t^2*x^2/2! + (t + t^3)*x^3/3! + ....
Hockey-stick recurrence: T(n+1,k+1) = Sum_{i = 0..floor((n-k)/2)} binomial(n,2*i)*T(n-2*i,k).
Recurrence equation for the row polynomials R(n,t):
R(n+1,t) = t*Sum_{k = 0..floor(n/2)} binomial(n,2*k)*R(n-2*k,t) with R(0,t) = 1. (End)

A008958 Triangle of central factorial numbers 4^k T(2n+1, 2n+1-2k).

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 35, 91, 1, 1, 84, 966, 820, 1, 1, 165, 5082, 24970, 7381, 1, 1, 286, 18447, 273988, 631631, 66430, 1, 1, 455, 53053, 1768195, 14057043, 15857205, 597871, 1, 1, 680, 129948, 8187608, 157280838, 704652312, 397027996, 5380840, 1

Offset: 0

N. J. A. Sloane

			From _Wesley Transue_, Jan 21 2012: (Start)
Triangle begins:
  1;
  1,   1;
  1,  10,      1;
  1,  35,     91,       1;
  1,  84,    966,     820,         1;
  1, 165,   5082,   24970,      7381,         1;
  1, 286,  18447,  273988,    631631,     66430,         1;
  1, 455,  53053, 1768195,  14057043,  15857205,    597871,       1;
  1, 680, 129948, 8187608, 157280838, 704652312, 397027996, 5380840, 1;
(End)

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

Robert James Purser, Mobius Net Cubed-Sphere Gnomonic Grids, U.S. Department of Commerce, National Oceanic and Atmospheric Administration, National Weather Service, National Centers for Environmental Protection, 2018.

Cf. A008955-A008957, A036969.

Columns include A000447. Right-hand columns include A002452, A002453.

Flatten[Table[Sum[(-1)^(q+1) 4^(p-n) (2p+2q-2n-1)^(2n+1)/((2n+1-2p-q)! q!), {q, 0, n-p}], {n, 0, 8}, {p, 0, n}]] (* Wesley Transue, Jan 21 2012 *)

G.f. of i-th right-hand column is x/Product_{j=1..i+1} (1 - (2j-1)^2*x).

More terms from Vladeta Jovovic, Apr 16 2000

A160562 Triangle of scaled central factorial numbers, T(n,k) = A008958(n,n-k).

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 91, 35, 1, 1, 820, 966, 84, 1, 1, 7381, 24970, 5082, 165, 1, 1, 66430, 631631, 273988, 18447, 286, 1, 1, 597871, 15857205, 14057043, 1768195, 53053, 455, 1, 1, 5380840, 397027996, 704652312, 157280838, 8187608, 129948, 680, 1

Offset: 0

Jonathan Vos Post, May 19 2009

This is table 4 on page 12 of Gelineau and Zeng, read downwards by columns.
Reversing rows gives A008958.
Apparently the table can also be obtained by deleting each second row and column of A136630.

			Triangle starts:
  1;
  1,     1;
  1,    10,      1;
  1,    91,     35,      1;
  1,   820,    966,     84,     1;
  1,  7381,  24970,   5082,   165,   1;
  1, 66430, 631631, 273988, 18447, 286, 1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows n = 0..150, flattened)
Qi Fang, Ya-Nan Feng, and Shi-Mei Ma, Alternating runs of permutations and the central factorial numbers, arXiv:2202.13978 [math.CO], 2022.
Yoann Gelineau and Jiang Zeng, Combinatorial Interpretations of the Jacobi-Stirling Numbers, arXiv:0905.2899 [math.CO], May 18 2009.

Cf. A002452 (column k=1), A002453 (column k=2), A000447 (right column k=n-1), A185375 (right column k=n-2).

Cf. A001819, A008275, A008277, A008958, A039599, A136630.

A160562 := proc(n,k) npr := 2*n+1 ; kpr := 2*k+1 ; sinh(t*sinh(x)) ; npr!*coeftayl(%,x=0,npr) ; coeftayl(%,t=0,kpr) ; end: seq(seq(A160562(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Sep 09 2009

T[n_, k_] := Sum[(-1)^(k - m)*(2m + 1)^(2n + 1)*Binomial[2k, k + m]/(k + m + 1), {m, 0, k}]/(4^k*(2k)!);
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 22 2017 *)

T(n,k) = (1/((2*k)!*4^k)) * Sum_{m=0..k} (-1)^(k-m)*A039599(k,m)*(2*m+1)^(2*n). - Werner Schulte, Nov 01 2015

T(n,k) = ((-1)^(n-k)*(2*n+1)!/(2*k+1)!) * [x^(2*n+1)]sin(x)^(2*k+1) = ((2*n+1)!/(2*k+1)!) * [x^(2*n+1)]sinh(x)^(2*k+1). Note that sin(x)^(2*k+1) = (Sum_{i=0..k} (-1)^i*binomial(2*k+1,k-i)*sin((2*i+1)*x))/(2^(2*k)). - Jianing Song, Oct 29 2023

More terms from R. J. Mathar, Sep 09 2009

A166984 a(n) = 20a(n-1) - 64a(n-2) for n > 1; a(0) = 1, a(1) = 20.

Original entry on oeis.org

1, 20, 336, 5440, 87296, 1397760, 22368256, 357908480, 5726601216, 91625881600, 1466015154176, 23456246661120, 375299963355136, 6004799480791040, 96076791961092096, 1537228672451215360, 24595658763514413056, 393530540233410478080, 6296488643803287126016

Offset: 0

Klaus Brockhaus, Oct 26 2009

Partial sums of A166965.

First differences of A006105. - Klaus Purath, Oct 15 2020

Seiichi Manyama, Table of n, a(n) for n = 0..830 (terms 0..200 from Vincenzo Librandi)
E. Saltürk and I. Siap, Generalized Gaussian Numbers Related to Linear Codes over Galois Rings, European Journal of Pure and Applied Mathematics, Vol. 5, No. 2, 2012, 250-259; ISSN 1307-5543. - From _N. J. A. Sloane_, Oct 23 2012
Index entries for linear recurrences with constant coefficients, signature (20,-64).

Cf. A000302, A002450, A002452, A002453, A006105, A079598, A083584.

Cf. A115490, A166914, A166917, A166927, A166965, A307695.

[n le 2 select 19*n-18 else 20*Self(n-1)-64*Self(n-2): n in [1..17] ];

LinearRecurrence[{20,-64},{1,20},30] (* Harvey P. Dale, Jul 04 2012 *)

a(n) = (4*16^n - 4^n)/3 \\ Charles R Greathouse IV, Jun 21 2022

A166984=BinaryRecurrenceSequence(20,-64,1,20)
[A166984(n) for n in range(31)] # G. C. Greubel, Oct 02 2024

a(n) = (4*16^n - 4^n)/3.
G.f.: 1/((1-4*x)*(1-16*x)).
Limit_{n -> oo} a(n)/a(n-1) = 16.
a(n) = A115490(n+1)/3.
Sum_{n>=0} a(n) x^(2*n+4)/(2*n+4)! = ( sinh(x) )^4/4!. - Robert A. Russell, Apr 03 2013
From Klaus Purath, Oct 15 2020: (Start)
a(n) = A002450(n+1)*(A002450(n+2) - A002450(n))/5.
a(n) = (A083584(n+1)^2 - A083584(n)^2)/80. (End)
a(n) = (A079598(n) - A000302(n))/24. - César Aguilera, Jun 21 2022
a(n) = 16*a(n-1) + 4^n with a(0) = 1. - Nadia Lafreniere, Aug 08 2022
E.g.f.: (4/3)*exp(10*x)*sinh(6*x + log(2)). - G. C. Greubel, Oct 02 2024

A381512 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = (2n+k)!/k! [x^(2*n+k)] sinh(x)^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 10, 16, 1, 0, 1, 20, 91, 64, 1, 0, 1, 35, 336, 820, 256, 1, 0, 1, 56, 966, 5440, 7381, 1024, 1, 0, 1, 84, 2352, 24970, 87296, 66430, 4096, 1, 0, 1, 120, 5082, 90112, 631631, 1397760, 597871, 16384, 1, 0, 1, 165, 10032, 273988, 3331328, 15857205, 22368256, 5380840, 65536, 1, 0

Offset: 0

Seiichi Manyama, May 11 2025

			Square array begins:
  1, 1,    1,     1,       1,        1, ...
  0, 1,    4,    10,      20,       35, ...
  0, 1,   16,    91,     336,      966, ...
  0, 1,   64,   820,    5440,    24970, ...
  0, 1,  256,  7381,   87296,   631631, ...
  0, 1, 1024, 66430, 1397760, 15857205, ...

Columns k=0..7 give A000007, A000012, A000302, A002452(n+1), A166984, A002453, 4^n * A002451(n), A381513.
Main diagonal gives A383837.
Cf. A136630, A160562, A286899.

a(n, k) = (2*n+k)!/k!*polcoef(sinh(x+x*O(x^(2*n+k)))^k, 2*n+k);

G.f. of column k: 1/Product_{j=0..floor(k/2)} (1 - (k-2*j)^2*x).
A(n,k) = k^2 * A(n-1,k) + A(n,k-2) for k > 1.
A(n,k) = (1/(2^k*k!)) * Sum_{j=0..k} (-1)^j * (k-2*j)^(2*n+k) * binomial(k,j).

cp's OEIS Frontend

A136630 Triangular array: T(n,k) counts the partitions of the set [n] into k odd sized blocks.

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A008958 Triangle of central factorial numbers 4^k T(2n+1, 2n+1-2k).

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A160562 Triangle of scaled central factorial numbers, T(n,k) = A008958(n,n-k).

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A166984 a(n) = 20*a(n-1) - 64*a(n-2) for n > 1; a(0) = 1, a(1) = 20.

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A381512 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = (2*n+k)!/k! * [x^(2*n+k)] sinh(x)^k.

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A166984 a(n) = 20a(n-1) - 64a(n-2) for n > 1; a(0) = 1, a(1) = 20.

A381512 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = (2n+k)!/k! [x^(2*n+k)] sinh(x)^k.