A008955 -id:A008955 - cp's OEIS frontend

A162440 The pg(n) sequence that is associated with the Eta triangle A160464.

2, 16, 144, 4608, 115200, 4147200, 203212800, 26011238400, 2106910310400, 210691031040000, 25493614755840000, 3671080524840960000, 620412608698122240000, 121600871304831959040000

Offset: 2

Johannes W. Meijer, Jul 06 2009

The EG1 matrix coefficients are defined by EG1[2m-1,1] = 2*eta(2m-1) and the recurrence relation EG1[2m-1,n] = EG1[2m-1,n-1] - EG1[2m-3,n-1]/(n-1)^2 with m = .., -2, -1, 0, 1, 2, ... and n = 1, 2, 3, ... . As usual, eta(m) = (1-2^(1-m))*zeta(m) with eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta function. For the EG2 matrix, the even counterpart of the EG1 matrix, see A008955.
The coefficients in the columns of the EG1 matrix, for m >= 1 and n >= 2, can be generated with GFE(z;n) = ((-1)^(n-1)*r(n)*CFN1(z,n)*GFE(z;n=1) + ETA(z,n))/pg(n) for n >= 2.
The CFN1(z,n) polynomials depend on the central factorial numbers A008955 and the ETA(z,n) are the Eta polynomials which led to the Eta triangle, see for both A160464.
The pg(n) sequence can be generated with the first Maple program and the EG1[2m-1,n] matrix coefficients can be generated with the second Maple program.
The EG1 matrix is related to the ES1 matrix, see A160464 and the formulas below.

			The first few generating functions GFE(z;n) are:
GFE(z;n=2) = ((-1)*2*(z^2 - 1)*GFE(z;n=1) + (-1))/2,
GFE(z;n=3) = ((+1)*4*(z^4 - 5*z^2 + 4)*GFE(z;n=1) + (-11 + 2*z^2))/16,
GFE(z;n=4) = ((-1)*4*(z^6-14*z^4+49*z^2-36)*GFE(z;n=1) + (-114+29*z^2-2*z^4))/144.

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 23, pp. 811-812.

The ETA(z, n) polynomials and the ES1 matrix lead to the Eta triangle A160464.
The CFN1(z, n), the t1(n, m) and the EG2 matrix lead to A008955.
The EG1[ -1, n] equal (1/2)*A001803(n-1)/A046161(n-1).
The r(n) sequence equals A062383(n) (n>=1).
The e(n) sequence equals A029837(n) (n>=1).
Cf. A160473 (p(n) sequence).
Cf. A162443 (BG1 matrix), A162446 (ZG1 matrix) and A162448 (LG1 matrix).

nmax := 16; seq((n-1)!^2*2^floor(ln(n-1)/ln(2)+1), n=2..nmax);
# End program 1
nmax1 := 5; coln := 4; mmax1 := nmax1: for n from 0 to nmax1 do t1(n, 0) := 1 end do: for n from 0 to nmax1 do t1(n, n) := (n!)^2 end do: for n from 1 to nmax1 do for m from 1 to n-1 do t1(n, m) := t1(n-1, m-1)*n^2 + t1(n-1, m) end do: end do: for m from 1 to mmax1 do EG1[1-2*m, 1] := evalf((2^(2*m)-1)* bernoulli(2*m)/(m)) od: EG1[1, 1] := evalf(2*ln(2)): for m from 2 to mmax1 do EG1[2*m-1, 1] := evalf(2*(1-2^(1-(2*m-1))) * Zeta(2*m-1)) od: for m from -mmax1+coln to mmax1 do EG1[2*m-1, coln]:= (-1)^(coln+1)*sum((-1)^k*t1(coln-1, k) * EG1[1-2*coln+2*m+2*k, 1], k=0..coln-1)/(coln-1)!^2 od;
# End program 2 (Edited by Johannes W. Meijer, Sep 21 2012)

pg(n) = (n-1)!^2*2^floor(log(n-1)/log(2)+1) for n >= 2.
r(n) = 2^e(n) = 2^floor(log(n-1)/log(2)+1) for n >= 2.
EG1[ -1,n] = 2^(1-2*n)*(2*n-1)!/((n-1)!^2) for n >= 1.
GFE(z;n) = sum (EG1[2*m-1,n]*z^(2*m-2), m=1..infinity).
GFE(z;n) = (1-z^2/(n-1)^2)*GFE(z;n-1)-EG1[ -1,n-1]/(n-1)^2 for n = >2. with GFE(z;n=1) = 2*log(2)-Psi(z)-Psi(-z)+Psi(z/2)+Psi(-z/2) and Psi(z) is the digamma function.
EG1[2m-1,n] = (2*2^(1-2*n)*(2*n-1)!/((n-1)!^2)) * ES1[2m-1,n].

A162446 Numerators of the column sums of the ZG1 matrix.

Original entry on oeis.org

-13, 401, -68323, 2067169, -91473331, 250738892357, -12072244190753, 105796895635531, -29605311573467996893, 9784971385947359480303, -5408317625058335310276319, 2111561851139130085557412009

Offset: 2

Johannes W. Meijer, Jul 06 2009

The ZG1 matrix coefficients are defined by ZG1[2m-1,1] = 2*zeta(2m-1) for m = 2, 3, .. , and the recurrence relation ZG1[2m-1,n] = (ZG1[2m-3,n-1] - (n-1)^2*ZG1[2m-1,n-1])/(n*(n-1)) with m = .. , -2, -1, 0, 1, 2, .. and n = 1, 2, 3, .. , under the condition that n <= (m-1). As usual zeta(m) is the Riemann zeta function. For the ZG2 matrix, the even counterpart of the ZG1 matrix, see A008955.
These two formulas enable us to determine the values of the ZG1[2*m-1,n] coefficients, with m all integers and n all positive integers, but not for all. If we choose, somewhat but not entirely arbitrarily, ZG1[1,1] = 2*gamma, with gamma the Euler-Mascheroni constant, we can determine them all.
The coefficients in the columns of the ZG1 matrix, for m >= 1 and n >= 2, can be generated with GFZ(z;n) = (hg(n)*CFN1(z;n)*GFZ(z;n=1) + ZETA(z;n))/pg(n) with pg(n) = 6*(n-1)!* (n)!*A160476(n) and hg(n) = 6*A160476(n). For the CFN1(z;n) and the ZETA(z;n) polynomials see A160474.
The column sums cs(n) = sum(ZG1[2*m-1,n], m = 1 .. infinity), for n >= 2, of the ZG1 matrix can be determined with the first Maple program. In this program we have made use of the remarkable fact that if we take ZGx[2*m-1,n] = 2, for m >= 1, and ZGx[ -1,n] = ZG1[ -1,n] and assume that the recurrence relation remains the same we find that the column sums of this new matrix converge to the same values as the original cs(n).
The ZG1[2*m-1,n] matrix coefficients can be generated with the second Maple program.
The ZG1 matrix is related to the ZS1 matrix, see A160474 and the formulas below.

			The first few generating functions GFZ(z;n) are:
GFZ(z;2) = (6*(1*z^2-1)*GFZ(z;1) + (-1))/12
GFZ(z;3) = (60*(z^4-5*z^2+4)*GFZ(z;1) + (51-10*z^2))/720
GFZ(z;4) = (1260*(z^6-14*z^4+49*z^2-36)*GFZ(z;1) + (-10594+2961*z^2-210*z^4))/181440

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 23, pp. 811-812.

See A162447 for the denominators of the column sums.
The pg(n) and hg(n) sequences lead to A160476.
The ZG1[ -1, n] coefficients lead to A000984, A002195 and A002196.
The ZETA(z, n) polynomials and the ZS1 matrix lead to the Zeta triangle A160474.
The CFN1(z, n), the cfn1(n, k) and the ZG2 matrix lead to A008955.
The b(n) sequence equals A001790(n)/ A120777(n-1) for n >= 1.
Cf. A001620 (gamma) and A010790 (n!*(n+1)!).
Cf. A162440 (EG1 matrix), A162443 (BG1 matrix) and A162448 (LG1 matrix)

nmax := 13; mmax := nmax: with(combinat): cfn1 := proc(n, k): sum((-1)^j1*stirling1(n+1, n+1-k+j1)*stirling1(n+1, n+1-k-j1), j1=-k..k) end proc: Omega(0):=1: for n from 1 to nmax do Omega(n) := (sum((-1)^(k1+n+1)*(bernoulli(2*k1)/(2*k1))*cfn1(n-1, n-k1), k1=1..n))/(2*n-1)! od: for n from 1 to nmax do ZG1[ -1, n] := binomial(2*n, n)*Omega(n) od: for n from 1 to nmax do ZGx[ -1, n] := ZG1[ -1, n] od: for m from 1 to mmax do ZGx[2*m-1, 1] := 2 od: for n from 2 to nmax do for m from 1 to mmax do ZGx[2*m-1, n] := (((ZGx[2*m-3, n-1]-(n-1)^2*ZGx[2*m-1, n-1])/(n*(n-1)))) od; s(n) := 0: for m from 1 to mmax do s(n) := s(n) + ZGx[2*m-1, n] od: od: seq(s(n), n=2..nmax);
# End program 1
nmax1 := 5; ncol := 3; Digits := 20: mmax1 := nmax1: with(combinat): cfn1 := proc(n, k): sum((-1)^j1*stirling1(n+1, n+1-k+j1)*stirling1(n+1, n+1-k-j1), j1=-k..k) end proc: ZG1[1, 1] := evalf(2*gamma): for m from 1 to mmax1 do ZG1[1-2*m, 1] := -bernoulli(2*m)/m od: for m from 2 to mmax1 do ZG1[2*m-1, 1] := evalf(2*Zeta(2*m-1)) od: for n from 1 to nmax1 do for m from -mmax1 to mmax1 do ZG1[2*m-1, n] := sum((-1)^(k1+1)*cfn1(n-1, k1-1)*ZG1[2*m-(2*n-2*k1+1), 1] /((n-1)!*(n)!), k1=1..n) od; od; for m from -mmax1+ncol to mmax1 do ZG1[2*m-1, ncol] := ZG1[2*m-1, ncol] od;
# End program 2
# Maple programs edited by Johannes W. Meijer, Sep 25 2012

a(n) = numer(cs(n)) and denom(cs(n)) = A162447(n).
with cs(n) = sum(ZG1[2*m-1,n], m = 1 .. infinity) for n >= 2.
GFZ(z;n) = sum( ZG1[2*m-1,n]*z^(2*m-2),m=1..infinity)
GFZ(z;n) = ZG1[ -1,n-1]/(n*(n-1))+(z^2-(n-1)^2)*GFZ(z;n-1)/(n*(n-1)) for n >= 2 with GFZ(z;n=1) = -Psi(1+z) - Psi(1-z).
ZG1[ -1,n] = binomial(2*n,n)*Omega[n] = A000984(n)*A002195(n)/A002196(n).
ZG1[2*m-1,n] = b(n)*ZS1[2*m-1,n] with b(n) = binomial(2*n,n)/2^(2*n-1) for n >= 1.

A160479 The ZL(n) sequence of the Zeta and Lambda triangles A160474 and A160487.

Original entry on oeis.org

10, 21, 2, 11, 13, 1, 34, 57, 5, 23, 1, 1, 29, 31, 2, 1, 37, 1, 41, 301, 1, 47, 1, 1, 53, 3, 1, 59, 61, 1, 2, 67, 1, 71, 73, 1, 1, 79, 1, 83, 1, 1, 89, 1, 1, 1, 97, 1, 505, 103, 1, 107, 109, 11, 113, 1, 1, 1, 1, 1, 1, 127, 2, 131

Offset: 3

Johannes W. Meijer, May 24 2009

The rather strange ZL(n) sequence rules both the Zeta and Lambda triangles.
The Zeta triangle led to the first and the Lambda triangle to the second Maple algorithm.
The first ZL(n) formula is a conjecture. This formula links the ZL(n) to the prime numbers A000040; see A217983, A128060, A130290 and the third Maple program.

Cf. A160474 and A160487.
The cnf1(n, k) are the central factorial numbers A008955.
The cnf2(n, k) are the central factorial numbers A008956.

nmax := 65; for n from 0 to nmax do cfn1(n, 0):=1: cfn1(n, n):=(n!)^2 end do: for n from 1 to nmax do for k from 1 to n-1 do cfn1(n, k) := cfn1(n-1, k-1)*n^2 + cfn1(n-1, k) end do: end do: Omega(0) := 1: for n from 1 to nmax do Omega(n) := (sum((-1)^(k1+n+1)*(bernoulli(2*k1)/(2*k1))*cfn1(n-1, n-k1), k1=1..n))/(2*n-1)! end do: for n from 1 to nmax do d(n) := 2^(2*n-1)*Omega(n) end do: for n from 1 to nmax do b(n) := 4^(-n)*(2*n+1)*n*denom(Omega(n)) end do: c(1) := b(1): for n from 1 to nmax-1 do c(n+1) := lcm(c(n)*(n+1)*(2*n+3)/2, b(n+1)) end do: for n from 1 to nmax do cm(n) := c(n)*(1/6)* 4^n/(2*n+1)! end do: for n from 3 to nmax+1 do ZL(n):=cm(n-1)/cm(n-2) end do: seq(ZL(n), n=3..nmax+1);
# End program 1 (program edited by Johannes W. Meijer, Oct 25 2012)
nmax1 := nmax; for n from 0 to nmax1 do cfn2(n, 0) :=1: cfn2(n, n) := (doublefactorial(2*n-1))^2 od: for n from 1 to nmax1 do for k from 1 to n-1 do cfn2(n, k) := (2*n-1)^2*cfn2(n-1, k-1) + cfn2(n-1, k) od: od: for n from 1 to nmax1 do Delta(n-1) := sum((1-2^(2*k1-1))* (-1)^(n+1)*(-bernoulli(2*k1)/(2*k1))*(-1)^(k1+n)*cfn2(n-1,n-k1), k1=1..n) /(2*4^(n-1)*(2*n-1)!) end do: for n from 1 to nmax1 do b(n) := (2*n)*(2*n-1)*denom(Delta(n-1))/ (2^(2*n)*(2*n-1)) end do: c(1) := b(1): for n from 1 to nmax1-1 do c(n+1) := lcm(c(n)*(2*n+2)* (2*n+1), b(n+1)) end do: for n from 1 to nmax1 do cm(n) := c(n)/(6*(2*n)!) end do: for n from 3 to nmax1+1 do ZL(n) := cm(n-1)/cm(n-2) end do: seq(ZL(n), n=3..nmax1+1);
# End program 2 (program edited by Johannes W. Meijer, Sep 20 2012)
nmax2 := nmax: A000040 := proc(n): ithprime(n) end: A130290 := proc(n): if n =1 then 1 else (A000040(n)-1)/2 fi: end: A128060 := proc(n) local n1: n1:=2*n-1: if type(n1, prime) then A128060(n) := 1 else A128060(n) := n1 fi: end: for n from 1 to nmax2 do A217983(n) := 1 od: for n from 1 to nmax2 do for n1 from 1 to floor(log[A000040(n)](nmax2)) do A217983(A130290(n) * A000040(n)^n1) := A000040(n) od: od: ZL := proc(n): (2*n-1)*(A217983(n-1)/A128060(n)) end: seq(ZL(n), n=3..nmax2+1);
# End program 3 (program added by Johannes W. Meijer, Oct 25 2012)

ZL(n) = (2*n-1) * (A217983(n-1)/A128060(n)) for n >= 3.
ZL(n) = ZETA(n, m)/(ZETA(n-1, m-1) - (n-1)^2 * ZETA(n-1, m)), see A160474.
ZL(n) = LAMBDA(n, m)/(LAMBDA(n-1, m-1) - (2*n-3)^2 * LAMBDA(n-1, m)), see A160487.
ZL(n) = A160476(n)/A160476(n-1).

Comments, formulas and third Maple program added by Johannes W. Meijer, Oct 25 2012

A163927 Numerators of the higher order exponential integral constants alpha(k,4).

Original entry on oeis.org

1, 49, 1897, 69553, 2515513, 90663937, 3264855049, 117543378001, 4231639039705, 152339702576545, 5484235568128681, 197432536935184369, 7107571838026381177, 255872590744254526273, 9211413307971174616393

Offset: 0

Johannes W. Meijer and Nico Baken, Aug 13 2009, Aug 17 2009

The higher order exponential integrals, see A163931, are defined by E(x,m,n) = x^(n-1)*Integral_{t>=x} E(t,m-1,n)/t^n for m >= 1 and n >= 1, with E(x,m=0,n) = exp(-x).
The series expansions of the higher order exponential integrals are dominated by the alpha(k,n) and the gamma(k,n) constants, see A090998.
The first Maple program uses the alpha(k,n) formula and the second the GF(z,n) to generate the alpha(k,n) coefficients in each column.
Appears to equal the numerator of the multiple harmonic (star) sum Sum_{1 <= k_1 <= ... <= k_n <= 3} 1/(k_1^2*...*k_n^2). If true, then a(n) = numerator( 3/2 - 3/(5*4^n) + 1/(10*9^n) ). - Peter Bala, Jan 31 2019

			a(k=0,n=4) = 1, a(k=1,4) = 49/36, a(k=2,4) = 1897/1296, a(k=3,4) = 69553/46656.

J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No. 3, April 1987, pp. 209-211.

Cf. A163931 (E(x,m,n)), A090998 (gamma(k,n)).
Cf. A163928 y A163929.
a(k,1) = A000007(k)
a(k,2) = A000012(k) = 1^k.
a(k,3) = A002450(k+1)/A000302(k) with A000302(k) = 4^k.
a(k,4) = A163927(k)/A009980(k) with A009980(k) = 36^k.
The GF(z,n) lead to A008955.
The denominators of a(1,n), n >= 2, lead to A007407.

coln := 4; nmax := 15; kmax := nmax: k:=0: for n from 1 to nmax do alpha(k, n) := 1 od: for k from 1 to kmax do for n from 1 to nmax do alpha(k, n) := (1/k)*sum(sum(p^(-2*(k-i)), p=0..n-1)*alpha(i, n), i=0..k-1) od; od: seq(alpha(k, coln), k=0..nmax-1);
# End program 1
coln:=4; nmax1 := 16; for n from 0 to nmax1 do A008955(n, 0):=1 end do: for n from 0 to nmax1 do A008955(n, n) := (n!)^2 end do: for n from 1 to nmax1 do for m from 1 to n-1 do A008955(n, m) := A008955(n-1, m-1)*n^2 + A008955(n-1, m) end do: end do: m:=coln-1: f(m):=0: for n from 0 to m do f(m) := f(m) + (-1)^(n + m)*A008955(m, n)*z^(2*m-2*n) od: GF(z,coln) := m!^2/f(m): GF(z,coln):=series(GF(z,coln), z, nmax1);
# End program 2

alpha(k,n) = (1/k) * Sum_{i=0..k-1} (Sum_{p=0..n-1}(p^(2*i-2*k))*alpha(i, n)) with alpha(0,n) = 1, k >= 0 and n >= 1.
alpha(k,n) = alpha(k,n+1) -alpha(k-1,n+1)/n^2.
GF(z,n) = product((1-(z/k)^2)^(-1), k = 1..n-1) = (Pi*z/sin(Pi*z))/(Beta(n+z,n-z)/Beta(n,n)).

A269944 Triangle read by rows, Stirling cycle 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) + (n-1)^2*T(n-1, k), for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 5, 1, 0, 36, 49, 14, 1, 0, 576, 820, 273, 30, 1, 0, 14400, 21076, 7645, 1023, 55, 1, 0, 518400, 773136, 296296, 44473, 3003, 91, 1, 0, 25401600, 38402064, 15291640, 2475473, 191620, 7462, 140, 1

Offset: 0

Peter Luschny, Mar 22 2016

Also known as central factorial numbers |t(2*n, 2*k)| (cf. A008955).

The analog for the Stirling set numbers is A269945.

			Triangle starts:
  [1]
  [0,     1]
  [0,     1,     1]
  [0,     4,     5,    1]
  [0,    36,    49,   14,    1]
  [0,   576,   820,  273,   30,  1]
  [0, 14400, 21076, 7645, 1023, 55, 1]

Peter Luschny, The P-transform.
Peter Luschny, The Partition Transform -- A SageMath Jupyter Notebook.
B. K. Miceli, Two q-Analogues of Poly-Stirling Numbers, J. Integer Seq., 14 (2011), 11.9.6.

Variants: A204579 (signed, row 0 missing), A008955.
Cf. A007318 (order 0), A132393 (order 1), A269947 (order 3).
Cf. A000330 (subdiagonal), A001044 (column 1), A101686 (row sums), A269945 (Stirling set), A269941 (P-transform).

T := proc(n, k) option remember; if n=k then return 1 fi; if k<0 or k>n then return 0 fi; T(n-1, k-1)+(n-1)^2*T(n-1, k) end: seq(seq(T(n, k), k=0..n), n=0..8);
# Alternatively with the P-transform (cf. A269941):
A269944_row := n -> PTrans(n, n->`if`(n=1, 1, (n-1)^2/(n*(4*n-2))), (n,k)->(-1)^k*(2*n)!/(2*k)!): seq(print(A269944_row(n)), n=0..8);
# From Peter Luschny, Feb 29 2024: (Start)
# Computed as the coefficients of polynomials:
P := (x, n) -> local j; mul((j - x)*(j + x), j = 0..n-1):
T := (n, k) -> (-1)^k*coeff(P(x, n), x, 2*k):
for n from 0 to 6 do seq(T(n, k), k = 0..n) od;
# Alternative, using the exponential generating function:
egf := cosh(2*arcsin(sqrt(t)*x/2)/sqrt(t)):
ser := series(egf, x, 20): cx := n -> coeff(ser, x, 2*n):
Trow := n -> local k; seq((2*n)!*coeff(cx(n), t, n-k), k = 0..n):
seq(print(Trow(n)), n = 0..9);  # (End)
# Alternative, row polynomials:
rowpoly := n -> pochhammer(-sqrt(x), n) * pochhammer(sqrt(x), n):
row := n -> local k; seq((-1)^k*coeff(expand(rowpoly(n)), x, k), k = 0..n):
seq(print(row(n)), n = 0..6);  # Peter Luschny, Aug 03 2024

T[n_, n_] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n - 1, k - 1] + (n - 1)^2*T[n - 1, k]; T[, ] = 0; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
(* Jean-François Alcover, Jul 25 2019 *)

stircycle2 = lambda n: 1 if n == 1 else (n-1)^2/(n*(4*n-2))
norm = lambda n,k: (-1)^k*factorial(2*n)/factorial(2*k)
M = PtransMatrix(7, stircycle2, 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) = (n - 1)^2 / (n*(4*n - 2)). The P-transform is defined in the link. See the Sage and Maple implementations below.
T(n, 1) = ((n - 1)!)^2 for n >= 1 (cf. A001044).
T(n, n-1) = n*(n - 1)*(2*n - 1)/6 for n >= 1 (cf. A000330).
Row sums: Product_{k=1..n} ((k - 1)^2 + 1) for n >= 0 (cf. A101686).
From Fabián Pereyra, Apr 25 2022: (Start)
T(n,k) = (-1)^(n-k)*Sum_{j=2*k..2*n} Stirling1(2*n,j)*binomial(j,2*k)*(n-1)^(j-2*k).
T(n,k) = Sum_{j=0..2*k} (-1)^(j - k)*Stirling1(n, j)*Stirling1(n, 2*k - j). (End)
From Peter Luschny, Feb 29 2024: (Start)
T(n, k) = (-1)^k*[x^(2*k)] P(x, n) where P(x, n) = Product_{j=0..n-1} (j-x)*(j+x).
T(n, k) = (2*n)!*[t^(n-k)] [x^(2*n)] cosh(2*arcsin(sqrt(t)*x/2)/sqrt(t)). (End)
T(n, k) = (-1)^k*[x^k] Pochhammer(-sqrt(x), n) * Pochhammer(sqrt(x), n). - Peter Luschny, Aug 03 2024

A008957 Triangle of central factorial numbers T(2n,2n-2*k), k >= 0, n >= 1 (in Riordan's notation).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 14, 21, 1, 1, 30, 147, 85, 1, 1, 55, 627, 1408, 341, 1, 1, 91, 2002, 11440, 13013, 1365, 1, 1, 140, 5278, 61490, 196053, 118482, 5461, 1, 1, 204, 12138, 251498, 1733303, 3255330, 1071799, 21845, 1, 1, 285, 25194, 846260, 10787231

Offset: 1

N. J. A. Sloane

D. E. Knuth [1992] on page 10 gives the formula Sum n^(2m-1) = Sum_{k=1..m} (2k-1)! T(2m,2k) binomial(n+k, 2k) where T(m, k) is the central factorial number of the second kind. - Michael Somos, May 08 2018

			The triangle starts:
  1;
  1,  1;
  1,  5,    1;
  1, 14,   21,     1;
  1, 30,  147,    85,     1;
  1, 55,  627,  1408,   341,    1;
  1, 91, 2002, 11440, 13013, 1365, 1;

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217, Table 6.2(a).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.8.

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
F. Alayont and N. Krzywonos, Rook Polynomials in Three and Higher Dimensions, 2012. [From _N. J. A. Sloane_, Jan 02 2013]
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
D. E. Knuth, Johann Faulhaber and Sums of Powers, arXiv:math/9207222, Jul 1992. See bottom of page 10. [From _Michael Somos_, May 08 2018]
Petro Kolosov, Polynomial identities involving central factorial numbers, GitHub, 2024. See p. 6.

Row reversed version of A036969. (0,0)-based version: A269945.

Cf. A008955.

a008957 n k = a008957_tabl !! (n-1) (k-1)
a008957_row n = a008957_tabl !! (n-1)
a008957_tabl = map reverse a036969_tabl
-- 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; # Gives rows of triangle in reversed order

t[n_, n_] = t[n_, 1] = 1;
t[n_, k_] := t[n-1, k-1] + k^2 t[n-1, k];
Flatten[Table[t[n, k], {n, 1, 10}, {k, n, 1, -1}]][[1 ;; 50]] (* Jean-François Alcover, Jun 16 2011 *)

{T(n, k) = if( n<1 || k>n, 0, n==k || k==1, 1, T(n-1, k-1) + k^2 * T(n-1, k))}; \\ Michael Somos, May 08 2018

def A008957(n, k):
    m = n - k
    return 2*sum((-1)^(j+m)*(j+1)^(2*n)/(factorial(j+m+2)*factorial(m-j)) for j in (0..m))
for n in (1..7): print([A008957(n, k) for k in (1..n)]) # Peter Luschny, May 10 2018

From Michael Somos, May 08 2018: (Start)
T(n, k) = T(n-1, k-1) + k^2 * T(n-1, k), where T(n, n) = T(n, 1) = 1.
E.g.f.: x^2 * (cosh(sinh(y*x/2) / (x/2)) - 1) = (1*x^2)*y^2/2! + (1*x^2 + 1*x^4)*y^4/4! + (1*x^2 + 5*x^4 + 1*x^6)*y^6/6! + (1*x^2 + 14*x^4 + 21*x^6 + 1*x^8)*y^8/8! + ... (End)

More terms from Vladeta Jovovic, Apr 16 2000

A161742 Third left hand column of the RSEG2 triangle A161739.

Original entry on oeis.org

1, 4, 13, 30, -14, -504, 736, 44640, -104544, -10644480, 33246720, 5425056000, -20843695872, -5185511654400, 23457840537600, 8506857655296000, -44092609863966720, -22430879475779174400, 130748316971139072000

Offset: 2

Johannes W. Meijer & Nico Baken (n.h.g.baken(AT)tudelft.nl), Jun 18 2009

Equals third left hand column of A161739 (RSEG2 triangle).
Other left hand columns are A129825 and A161743.
A008955 is a central factorial number triangle.
A028246 is Worpitzky's triangle.
A001710 (n!/2!), A001715 (n!/3!), A001720 (n!/4!), A001725 (n!/5!), A001730 (n!/6!), A049388 (n!/7!), A049389 (n!/8!), A049398 (n!/9!), A051431 (n!/10!) appear in Maple program.

nmax:=21; for n from 0 to nmax do A008955(n,0):=1 end do: for n from 0 to nmax do A008955(n,n):=(n!)^2 end do: for n from 1 to nmax do for m from 1 to n-1 do A008955(n,m):= A008955(n-1,m-1)*n^2+A008955(n-1,m) end do: end do: for n from 1 to nmax do A028246(n,1):=1 od: for n from 1 to nmax do A028246(n,n):=(n-1)! od: for n from 3 to nmax do for m from 2 to n-1 do A028246(n,m):=m*A028246(n-1,m)+(m-1)*A028246(n-1,m-1) od: od: for n from 2 to nmax do a(n):=sum(((-1)^k/((k+1)!*(k+2)!)) *(n!)*A028246(n,k+2)* A008955(k+1,k),k=0..n-2) od: seq(a(n),n=2..nmax);

a(n) = sum(((-1)^k/((k+1)!*(k+2)!))*(n!)*A028246(n, k+2)*A008955(k+1, k), k=0..n-2)

A161743 Fourth left hand column of the RSEG2 triangle A161739.

Original entry on oeis.org

1, 10, 73, 425, 1561, -2856, -73520, 380160, 15376416, -117209664, -7506967104, 72162155520, 7045087741056, -80246202992640, -11448278791372800, 149576169325363200, 30017051616972275712, -440857664887810867200

Offset: 3

Johannes W. Meijer & Nico Baken (n.h.g.baken(AT)tudelft.nl), Jun 18 2009

Equals fourth left hand column of A161739 (RSEG2 triangle).
Other left hand columns are A129825 and A161742.
A008955 is a central factorial number triangle.
A028246 is Worpitzky's triangle.
A001710 (n!/2!), A001715 (n!/3!), A001720 (n!/4!), A001725 (n!/5!), A001730 (n!/6!), A049388 (n!/7!), A049389 (n!/8!), A049398 (n!/9!), A051431 (n!/10!) appear in Maple program.

nmax:=21; for n from 0 to nmax do A008955(n,0):=1 end do: for n from 0 to nmax do A008955(n,n):=(n!)^2 end do: for n from 1 to nmax do for m from 1 to n-1 do A008955(n,m):= A008955(n-1,m-1)*n^2+A008955(n-1,m) end do: end do: for n from 1 to nmax do A028246(n,1):=1 od: for n from 1 to nmax do A028246(n,n):=(n-1)! od: for n from 3 to nmax do for m from 2 to n-1 do A028246(n,m):=m*A028246(n-1,m)+(m-1)*A028246(n-1,m-1) od: od: for n from 3 to nmax do a(n) := sum(((-1)^k/((k+2)!*(k+3)!))*(n!)*A028246(n,k+3)* A008955(k+2,k), k=0..n-3) od: seq(a(n),n=3..nmax);

a(n) = sum(((-1)^k/((k+2)!*(k+3)!))*(n!)*A028246(n, k+3)*A008955(k+2, k), k = 0..n-3).

A182867 Triangle read by rows: row n gives coefficients in expansion of Product_{i=1..n} (x - (2i)^2), highest powers first.

Original entry on oeis.org

1, 1, -4, 1, -20, 64, 1, -56, 784, -2304, 1, -120, 4368, -52480, 147456, 1, -220, 16368, -489280, 5395456, -14745600, 1, -364, 48048, -2846272, 75851776, -791691264, 2123366400, 1, -560, 119392, -12263680, 633721088, -15658639360, 157294854144, -416179814400, 1, -816, 262752, -42828032, 3773223168, -177891237888, 4165906530304, -40683662475264, 106542032486400, 1, -1140, 527136, -127959680, 17649505536, -1400415544320, 61802667606016, -1390437378293760, 13288048674471936, -34519618525593600

Offset: 0

N. J. A. Sloane, Feb 01 2011

These are scaled central factorial numbers (see the discussion in the Comments section of A008955). The coefficients in the expansion of Product_{i=1..n} (x - i^2) give A008955, and the coefficients in the expansion of Product_{i=1..n} (x - (2i+1)^2) give A008956.

			Triangle begins:
 1
 1, -4
 1, -20, 64
 1, -56, 784, -2304
 1, -120, 4368, -52480, 147456
 1, -220, 16368, -489280, 5395456, -14745600
 1, -364, 48048, -2846272, 75851776, -791691264, 2123366400
 1, -560, 119392, -12263680, 633721088, -15658639360, 157294854144, -416179814400
 1, -816, 262752, -42828032, 3773223168, -177891237888, 4165906530304, -40683662475264, 106542032486400
 1, -1140, 527136, -127959680, 17649505536, -1400415544320, 61802667606016, -1390437378293760, 13288048674471936, -34519618525593600
...
For example, for n=2, (x-4)(x-16) = x^2 - 20x + 64 => [1, -20, 64].

T. L. Curtright, D. B. Fairlie, and C. K. Zachos, A compact formula for rotations as spin matrix polynomials, arXiv preprint arXiv:1402.3541 [math-ph], 2014.
T. L. Curtright and T. S. Van Kortryk, On Rotations as Spin Matrix Polynomials, arXiv:1408.0767 [math-ph], 2014.
T. L. Curtright, More on Rotations as Spin Matrix Polynomials, arXiv preprint arXiv:1506.04648 [math-ph], 2015.

Cf. A008955, A008956. This triangle is formed from the even-indexed rows of A182971 (the odd-indexed rows give A008956).

Cf. A160563.

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 10 do
t1:=eval(Q(2*n)); t1d:=degree(t1);
t12:=y^t1d*subs(x=1/y,t1); t2:=seriestolist(series(t12,y,20));
lprint(t2);
od:
# Using a bivariate generating function (adding a superdiagonal 1,0,0, ...):
gf := (t + sqrt(1 + t^2))^x:
ser := series(gf, t, 20): ct := n -> coeff(ser, t, n):
T := (n, k) -> n!*coeff(ct(n), x, n - k):
EvenPart := (T, len) -> local n, k;
seq(print(seq(T(n, k), k = 0..n, 2)), n = 0..2*len-1, 2):
EvenPart(T, 6);  # Peter Luschny, Mar 03 2024

Given a (0, 0)-based triangle U we call the triangle [U(n, k), k=0..n step 2, n=0..len step 2] the 'even subtriangle' of U. This triangle is the even subtriangle of U(n, k) = n! * [x^(n-k)] [t^n] (t + sqrt(1 + t^2))^x, albeit adding a superdiagonal 1, 0, 0, ... See A160563 for the odd subtriangle. - Peter Luschny, Mar 03 2024

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).

cp's OEIS Frontend

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A008957 Triangle of central factorial numbers T(2*n,2*n-2*k), k >= 0, n >= 1 (in Riordan's notation).

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A161742 Third left hand column of the RSEG2 triangle A161739.

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A008957 Triangle of central factorial numbers T(2n,2n-2*k), k >= 0, n >= 1 (in Riordan's notation).

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.