A121408 -id:A121408 - 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)

A385343 Exponential Riordan array (1, arcsin(x)).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 4, 0, 1, 0, 9, 0, 10, 0, 1, 0, 0, 64, 0, 20, 0, 1, 0, 225, 0, 259, 0, 35, 0, 1, 0, 0, 2304, 0, 784, 0, 56, 0, 1, 0, 11025, 0, 12916, 0, 1974, 0, 84, 0, 1, 0, 0, 147456, 0, 52480, 0, 4368, 0, 120, 0, 1, 0, 893025, 0, 1057221, 0, 172810, 0, 8778, 0, 165, 0, 1

Offset: 0

Seiichi Manyama, Jun 26 2025

			Triangle starts:
  1;
  0,   1;
  0,   0,  1;
  0,   1,  0,   1;
  0,   0,  4,   0,  1;
  0,   9,  0,  10,  0,  1;
  0,   0, 64,   0, 20,  0, 1;
  0, 225,  0, 259,  0, 35, 0, 1;

Essentialy same as A121408.
Row sums give A006228.
Cf. A091885, A136630, A185951.

T(n, k) = my(x='x+O('x^(n+1))); n!*polcoef(asin(x)^k/k!, n);

E.g.f. of column k (with leading zeros): arcsin(x)^k / k!

T(n,k) = A121408(n,k) for k > 0.

A182971 Triangle read by rows: coefficients in expansion of Q(n) = (x-n^2)(x-(n-2)^2)(x-(n-4)^2)...(x-(1 or 2)^2), highest powers first.

Original entry on oeis.org

1, 1, -1, 1, -4, 1, -10, 9, 1, -20, 64, 1, -35, 259, -225, 1, -56, 784, -2304, 1, -84, 1974, -12916, 11025, 1, -120, 4368, -52480, 147456, 1, -165, 8778, -172810, 1057221, -893025, 1, -220, 16368, -489280, 5395456, -14745600, 1, -286, 28743, -1234948, 21967231, -128816766, 108056025, 1, -364, 48048, -2846272, 75851776, -791691264, 2123366400

Offset: 0

N. J. A. Sloane, Feb 01 2011

These are scaled versions of the central factorial numbers in A008955 and A008956.
Even-indexed rows give A182867, odd-indexed rows give A008956.
A121408 is an unsigned and aerated version of the row reverse of this triangle. - Peter Bala, Aug 29 2012

			Triangle begins:
1
1, -1
1, -4
1, -10, 9
1, -20, 64
1, -35, 259, -225
1, -56, 784, -2304
1, -84, 1974, -12916, 11025
1, -120, 4368, -52480, 147456
1, -165, 8778, -172810, 1057221, -893025
1, -220, 16368, -489280, 5395456, -14745600
...
E.g. for n=5 Q(5) = (x-1^2)*(x-3^2)*(x-5^2) = x^3-35*x^2+259*x-225.

Even-indexed rows give A182867, odd-indexed rows give A008956.
Column 1,4,10,20, ... is A000292. The next two columns give A181888, A184878. The last diagonal is A184877.
Cf. A008955, A008956. A121408.

Q:= n -> if n mod 2 = 0 then sort(expand(mul(x-4*i^2,i=1..n/2)));
else sort(expand(mul(x-(2*i+1)^2,i=0..(n-1)/2))); fi;
for n from 0 to 12 do
t1:=eval(Q(n)); t1d:=degree(t1);
t12:=y^t1d*subs(x=1/y,t1); t2:=seriestolist(series(t12,y,20));
lprint(t2);
od:

For n even, let Q(n) = Product_{i=1..n/2} (x - (2*i)^2) and for n odd let Q(n) = Product_{i=0..(n-1)/2} (x - (2i+1)^2). n-th row of triangle gives coefficients in expansion of Q(n).

A204579 Triangle read by rows: matrix inverse of the central factorial numbers T(2n, 2k) (A036969).

Original entry on oeis.org

1, -1, 1, 4, -5, 1, -36, 49, -14, 1, 576, -820, 273, -30, 1, -14400, 21076, -7645, 1023, -55, 1, 518400, -773136, 296296, -44473, 3003, -91, 1, -25401600, 38402064, -15291640, 2475473, -191620, 7462, -140, 1, 1625702400, -2483133696, 1017067024, -173721912, 14739153, -669188, 16422, -204, 1

Offset: 1

M. F. Hasler, Feb 03 2012

This is a signed version of A008955 with rows in reverse order. - Peter Luschny, Feb 04 2012

			Triangle starts:
  [1]         1;
  [2]        -1,        1;
  [3]         4,       -5,         1;
  [4]       -36,       49,       -14,       1;
  [5]       576,     -820,       273,     -30,       1;
  [6]    -14400,    21076,     -7645,    1023,     -55,    1;
  [7]    518400,  -773136,    296296,  -44473,    3003,  -91,    1;
  [8] -25401600, 38402064, -15291640, 2475473, -191620, 7462, -140, 1;

José L. Cereceda, Sums of powers of integers and the sequence A304330, arXiv:2405.05268 [math.GM], 2024. See p. 6.
Mark W. Coffey and Matthew 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.

Cf. A036969, A008955, A008275, A121408, A001044 (column 1), A101686 (alternating row sums), A234324 (central terms).

# From Peter Luschny, Feb 29 2024: (Start)
ogf := n -> local j; z^2*mul(z^2 - j^2, j = 1..n-1):
Trow := n -> local k; seq(coeff(expand(ogf(n)), z, 2*k), k = 1..n):
# Alternative:
f := w -> (w^sqrt(t) + w^(-sqrt(t)))/2: egf := f((x/2 + sqrt(1 + (x/2)^2))^2):
ser := series(egf, x, 20): cx := n -> coeff(ser, x, 2*n):
Trow := n -> local k; seq((2*n)!*coeff(cx(n), t, k), k = 1..n):  # (End)
# Assuming offset 0:
rowpoly := n -> (-1)^n * pochhammer(1 - sqrt(x), n) * pochhammer(1 + sqrt(x), n):
row := n -> local k; seq(coeff(expand(rowpoly(n)), x, k), k = 0..n):
seq(print(row(n)), n = 0..7);  # Peter Luschny, Aug 03 2024

rows = 10;
t[n_, k_] := 2*Sum[j^(2*n)*(-1)^(k - j)/((k - j)!*(k + j)!), {j, 1, k}];
T = Table[t[n, k], {n, 1, rows}, {k, 1, rows}] // Inverse;
Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 14 2018 *)

select(concat(Vec(matrix(10,10,n,k,T(n,k)/*from A036969*/)~^-1)), x->x)

def A204579(n, k): return (-1)^(n-k)*A008955(n, n-k)
for n in (0..7): print([A204579(n, k) for k in (0..n)]) # Peter Luschny, Feb 05 2012

T(n, k) = (-1)^(n-k)*A008955(n, n-k). - Peter Luschny, Feb 05 2012
T(n, k) = Sum_{i=k-n..n-k} (-1)^(n-k+i)*s(n,k+i)*s(n,k-i) = Sum_{i=0..2*k} (-1)^(n+i)*s(n,i)*s(n,2*k-i), where s(n,k) are Stirling numbers of the first kind, A048994. - Mircea Merca, Apr 07 2012
From Peter Bala, Aug 29 2012: (Start)
T(n, k) = T(n-1, k-1) - (n-1)^2*T(n-1, k). (Recurrence equation.)
Let E(x) = cosh(sqrt(2*x)) = Sum_{n >= 0} x^n/{(2*n)!/2^n} and
L(x) = 2*{arcsinh(sqrt(x/2))}^2 = Sum_{n >=1} (-1)^n*(n-1)!^2*x^n/{(2*n)!/2^n}.
L(x) is the compositional inverse of E(x) - 1.
A generating function for the triangle is E(t*L(x)) = 1 + t*x + t*(-1 + t)*x^2/6 + t*(4 - 5*t + t^2)*x^3/90 + ..., where the sequence of denominators [1,1,6,90,...] is given by (2*n)!/2^n. Cf. A008275 with generating function exp(t*log(1+x)).
The e.g.f. is E(t*L(x^2/2)) = cosh(2*sqrt(t)*arcsinh(x/2)) = 1 + t*x^2/2! + t*(t-1)*x^4/4! + t*(t-1)*(t-4)*x^6/6! + .... (End)
From Peter Luschny, Feb 29 2024: (Start)
T(n, k) = [z^(2*k)] z^2*Product_{j=1..n-1} (z^2 - j^2).
T(n, k) = (2*n)! * [t^k] [x^(2*n)] (w^sqrt(t) + w^(-sqrt(t)))/2 where w = (x/2 + sqrt(1 + (x/2)^2))^2. (End)
T(n, k) = [x^k] (-1)^n * Pochhammer(1 - sqrt(x), n) * Pochhammer(1 + sqrt(x), n), assuming offset 0. - Peter Luschny, Aug 03 2024
Integral_{0..oo} x^s / (cosh(x))^(2*n) dx = (2^(2*n - s - 1) * s! * (-1)^(n-1)) / (2*n - 1)!)*Sum_{k=1..n} T(n,k)*DirichletEta(s - 2*k + 2). - Ammar Khatab, Apr 11 2025

Typo in data corrected by Peter Luschny, Feb 05 2012

A091885 Triangle T(n,k) defined by the generating function cosh(sqrt(y)arcsin(x)) + sqrt(y)sinh(sqrt(y)arcsin(x)) - 1 = Sum_{n>=1} Sum_{k=1..n} T(n,k)y^k *x^n/n!.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 9, 10, 1, 64, 20, 1, 225, 259, 35, 1, 2304, 784, 56, 1, 11025, 12916, 1974, 84, 1, 147456, 52480, 4368, 120, 1, 893025, 1057221, 172810, 8778, 165, 1, 14745600, 5395456, 489280, 16368, 220, 1, 108056025, 128816766, 21967231, 1234948, 28743

Offset: 1

Karol A. Penson, Feb 08 2004

Row sums are equal to A006228(n). This is sequence A121408 without the intertwining zeros. - Emeric Deutsch, Jul 28 2006

This number triangle corresponds to the coefficients of the polynomial of the denominator of Fourier cosine coefficients for functions of the form sin(x)^(2*k) for integer n. For example (k=5), evaluating Integral_{x=-Pi..Pi} cos(n*x)*sin(x)^10 dx, we have -7257600*sin(n*Pi)/(-14745600*n + 5395456*n^3 - 489280*n^5 + 16368*n^7 - 220*n^9 + n^11); note the sequence of the coefficients of the polynomial of the denominator: -14745600, 5395456, -489280, 16368, -220, 1. - John M. Campbell, May 28 2011

			Triangle starts:
    1;
    1;
    1,   1;
    4,   1;
    9,  10,   1;
   64,  20,   1;
  225, 259,  35,   1;

Cf. A006228.

Cf. A121408.

G:=cosh(sqrt(y)*arcsin(x))+sqrt(y)*sinh(sqrt(y)*arcsin(x))-1: Gser:=simplify(series(G,x=0,15)): for n from 1 to 13 do P[n]:=sort(expand(n!*coeff(Gser,x,n))) od: for n from 1 to 13 do seq(coeff(P[n],y,k),k=1..ceil(n/2)) od; # yields sequence in triangular form # Emeric Deutsch, Jul 28 2006

m = 14; (* number of rows *)
T = Rest /@ Rest[CoefficientList[#, y]& /@ (CoefficientList[Cosh[Sqrt[y]* ArcSin[x]] + Sqrt[y]*Sinh[Sqrt[y]*ArcSin[x]] - 1  + O[x]^(m + 1), x]* Range[0, m]! // Simplify[#, y > 0]&)];
Flatten[T] (* Jean-François Alcover, Sep 27 2021 *)

E.g.f.: cosh(sqrt(y)*arcsin(x))+sqrt(y)*sinh(sqrt(y)*arcsin(x))-1.

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004

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

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A385343 Exponential Riordan array (1, arcsin(x)).

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A182971 Triangle read by rows: coefficients in expansion of Q(n) = (x-n^2)*(x-(n-2)^2)*(x-(n-4)^2)*...*(x-(1 or 2)^2), highest powers first.

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A204579 Triangle read by rows: matrix inverse of the central factorial numbers T(2*n, 2*k) (A036969).

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A091885 Triangle T(n,k) defined by the generating function cosh(sqrt(y)*arcsin(x)) + sqrt(y)*sinh(sqrt(y)*arcsin(x)) - 1 = Sum_{n>=1} Sum_{k=1..n} T(n,k)*y^k *x^n/n!.

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A182971 Triangle read by rows: coefficients in expansion of Q(n) = (x-n^2)(x-(n-2)^2)(x-(n-4)^2)...(x-(1 or 2)^2), highest powers first.

A204579 Triangle read by rows: matrix inverse of the central factorial numbers T(2n, 2k) (A036969).

A091885 Triangle T(n,k) defined by the generating function cosh(sqrt(y)arcsin(x)) + sqrt(y)sinh(sqrt(y)arcsin(x)) - 1 = Sum_{n>=1} Sum_{k=1..n} T(n,k)y^k *x^n/n!.