A002866 -id:A002866 - cp's OEIS frontend

A178802 Multiply, cell by cell, sequence A048996 by A178801.

1, 1, 2, 2, 6, 12, 6, 24, 48, 24, 72, 24, 120, 240, 240, 360, 360, 480, 120, 720, 1440, 1440, 720, 2160, 4320, 720, 2880, 4320, 3600, 720, 5040, 10080, 10080, 10080, 15120, 30240, 15120, 15120, 20160, 60480, 20160, 25200, 50400, 30240, 5040, 40320, 80640, 80640, 80640, 40320, 120960

Offset: 0

Alford Arnold, Jun 15 2010

			A048996 begins 1,1,1,1,1,2,1, 1, 2, 1, 3, 1 ...
A178801 begins 1,1,2,2,6,6,6,24,24,24,24,24,...
therefore
a(n)    begins 1,1,2,2,6,12,6,...
with row sums 1, 1, 4, 24, 192, 1920, 23040, ... .
Triangle begins:
    1;
    1;
    2,    2;
    6,   12,    6;
   24,   48,   24,  72,   24;
  120,  240,  240, 360,  360,  480, 120;
  720, 1440, 1440, 720, 2160, 4320, 720, 2880, 4320, 3600, 720;
  ...

Cf. A000041, A000142, A048996, A178801, A002866 (row sums).

T(n,k) = A048996(n,k) * A178801(n,k) = A048996(n,k) * n!.

A274844 The inverse multinomial transform of A001818(n) = ((2*n-1)!!)^2.

Original entry on oeis.org

1, 8, 100, 1664, 34336, 843776, 24046912, 779780096, 28357004800, 1143189536768, 50612287301632, 2441525866790912, 127479926768287744, 7163315850315825152, 431046122080208896000, 27655699473265974050816, 1884658377677216933085184

Offset: 1

Johannes W. Meijer, Jul 27 2016

nonn

The inverse multinomial transform [IML] transforms an input sequence b(n) into the output sequence a(n). The IML transform inverses the effect of the multinomial transform [MNL], see A274760, and is related to the logarithmic transform, see A274805 and the first formula.
To preserve the identity MNL[IML[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL.
In the a(n) formulas, see the examples, the cumulant expansion numbers A127671 appear.
We observe that the inverse multinomial transform leaves the value of a(0) undefined.
The Maple programs can be used to generate the inverse multinomial transform of a sequence. The first program is derived from a formula given by Alois P. Heinz for the logarithmic transform, see the first formula and A001187. The second program uses the e.g.f. for multivariate row polynomials, see A127671 and the examples. The third program uses information about the inverse of the inverse of the multinomial transform, see A274760.
The IML transform of A001818(n) = ((2*n-1)!!)^2 leads quite unexpectedly to A005411(n), a sequence related to certain Feynman diagrams.
Some IML transform pairs, n >= 1: A000110(n) and 1/A000142(n-1); A137341(n) and A205543(n); A001044(n) and A003319(n+1); A005442(n) and A000204(n); A005443(n) and A001350(n); A007559(n) and A000244(n-1); A186685(n+1) and A131040(n-1); A061711(n) and A141151(n); A000246(n) and A000035(n); A001861(n) and A141044(n-1)/A001710(n-1); A002866(n) and A000225(n); A000262(n) and A000027(n).

			Some a(n) formulas, see A127671:
a(0) = undefined
a(1) = (1/0!) * (1*x(1))
a(2) = (1/1!) * (1*x(2) - x(1)^2)
a(3) = (1/2!) * (1*x(3) - 3*x(2)*x(1) + 2*x(1)^3)
a(4) = (1/3!) * (1*x(4) - 4*x(3)*x(1) - 3*x(2)^2 + 12*x(2)*x(1)^2 - 6*x(1)^4)
a(5) = (1/4!) * (1* x(5) - 5*x(4)*x(1) - 10*x(3)*x(2) + 20*x(3)*x(1)^2 + 30*x(2)^2*x(1) -60*x(2)*x(1)^3 + 24*x(1)^5)

Richard P. Feynman, QED, The strange theory of light and matter, 1985.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Logarithmic Transform.
Wikipedia, Feynman diagram

Cf. A274760, A274805, A127671, A001187, A001818, A000079, A005411.

nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: c:= proc(n) option remember; b(n) - add(k*binomial(n, k)*b(n-k)*c(k), k=1..n-1)/n end: a := proc(n): c(n)/(n-1)! end: seq(a(n), n=1..nmax); # End first IML program.
nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := log(1+add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n*coeff(t2, x, n) end: seq(a(n), n=1..nmax); # End second IML program.
nmax:=17: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(exp(add(t(n)*x^n/n, n=1..nmax)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(1):=b(1): t(1):= b(1): for n from 2 to nmax+1 do t(n) := solve(d(n)-b(n), t(n)): a(n):=t(n): od: seq(a(n), n=1..nmax); # End third IML program.

nMax = 22; b[n_] := ((2*n-1)!!)^2; c[n_] := c[n] = b[n] - Sum[k*Binomial[n, k]*b[n-k]*c[k], {k, 1, n-1}]/n; a[n_] := c[n]/(n-1)!; Table[a[n], {n, 1, nMax}] (* Jean-François Alcover, Feb 27 2017, translated from Maple *)

a(n) = c(n)/(n-1)! with c(n) = b(n) - Sum_{k=1..n-1}(k*binomial(n, k)*b(n-k)*c(k)), n >= 1 and a(0) = undefined, with b(n) = A001818(n) = ((2*n-1)!!)^2.

a(n) = A000079(n-1) * A005411(n), n >= 1.

A305578 a(n) = Sum_{k=0..n} binomial(n,k)k!!(n - k)!!.

Original entry on oeis.org

1, 2, 6, 18, 64, 230, 936, 3822, 17344, 78354, 389280, 1913010, 10267776, 54235350, 311348352, 1751907150, 10673326080, 63531238050, 408231498240, 2556121021650, 17236028160000, 113006008398150, 796296326031360, 5445783239554350, 39959419088977920, 284127133728611250

Offset: 0

Ilya Gutkovskiy, Jun 05 2018

nonn

Exponential convolution of A006882 with itself.

Alois P. Heinz, Table of n, a(n) for n = 0..730
Eric Weisstein's World of Mathematics, Double Factorial
Index entries for sequences related to factorial numbers

Cf. A000165, A001563, A002866, A006882, A305577.

a:= proc(n) option remember; `if`(n<4, [1, 2, 6, 18][n+1],
      3*n*a(n-2)-2*(n-3)*n*a(n-4))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 14 2018

Table[Sum[Binomial[n, k] k!! (n - k)!!, {k, 0, n}], {n, 0, 25}]
nmax = 25; CoefficientList[Series[(1 + x Exp[x^2/2] (1 + Sqrt[Pi/2] Erf[x/Sqrt[2]]))^2, {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: (1 + x*exp(x^2/2)*(1 + sqrt(Pi/2)*erf(x/sqrt(2))))^2.

A308543 Expansion of e.g.f. exp(2(exp(2x) - 1)).

Original entry on oeis.org

1, 4, 24, 176, 1504, 14528, 155520, 1819392, 23019008, 312413184, 4518705152, 69279690752, 1120856170496, 19062628335616, 339681346551808, 6323658075340800, 122680376836358144, 2474677219852288000, 51799971194270646272, 1123121391647711035392

Offset: 0

Ilya Gutkovskiy, Jun 06 2019

nonn

Cf. A000079, A000110, A001861, A002866, A004211, A055882.

nmax = 19; CoefficientList[Series[Exp[2 (Exp[2 x] - 1)], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[Sum[4^k x^k/Product[(1 - 2 j x), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = Sum[2^(k + 1) Binomial[n - 1, k - 1] a[n - k], {k, n}]; a[0] = 1; Table[a[n], {n, 0, 19}]
Table[2^n BellB[n, 2], {n, 0, 19}]

O.g.f.: Sum_{k>=0} 4^k*x^k / Product_{j=1..k} (1 - 2*j*x).
E.g.f.: exp(4*exp(x)*sinh(x)).
E.g.f.: g(g(x) - 1), where g(x) = e.g.f. of A000079 (powers of 2).
E.g.f.: f(x)^4, where f(x) = e.g.f. of A004211 (shifts one place left under 2nd-order binomial transform).
a(0) = 1; a(n) = Sum_{k=1..n} 2^(k+1)*binomial(n-1,k-1)*a(n-k).
a(n) = Sum_{k=0..n} 2^(n+k)*Stirling2(n,k).
a(n) = exp(-2) * Sum_{k>=0} 2^(n+k)*k^n/k!.
a(n) = 2^n * A001861(n).

A308878 Expansion of e.g.f. (1 - log(1 + x))/(1 - 2*log(1 + x)).

Original entry on oeis.org

1, 1, 3, 14, 86, 664, 6136, 66240, 816672, 11331552, 174662304, 2961774144, 54785368128, 1097882522112, 23693117756928, 547844658441216, 13511950038494208, 354086653712228352, 9824794572366544896, 287752569360558907392, 8871374335098501292032

Offset: 0

Ilya Gutkovskiy, Jun 29 2019

nonn

Inverse Stirling transform of A002866.

Cf. A002866, A008275, A011782, A050351, A088501, A306042, A308877.

nmax = 20; CoefficientList[Series[(1 - Log[1 + x])/(1 - 2 Log[1 + x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[StirlingS1[n, k] 2^(k - 1) k!, {k, 1, n}], {n, 1, 20}]]

a(0) = 1; a(n) = Sum_{k=1..n} Stirling1(n,k) * 2^(k-1) * k!.

a(n) ~ n! * exp(1/2) / (4 * (exp(1/2) - 1)^(n+1)). - Vaclav Kotesovec, Jun 29 2019

A317618 Expansion of e.g.f. sqrt((1 - x)/(1 - 3*x)).

Original entry on oeis.org

1, 1, 5, 39, 417, 5685, 94365, 1847475, 41686785, 1065288105, 30411314325, 959236098975, 33129890726625, 1243507150410525, 50401090111697325, 2193907232242600875, 102075654396429338625, 5055304328553234380625, 265522264682686831945125, 14742355948224269570580375

Offset: 0

Ilya Gutkovskiy, Aug 01 2018

nonn

Lah transform of A001147.

Robert Israel, Table of n, a(n) for n = 0..380
N. J. A. Sloane, Transforms

Cf. A000246, A001147, A002866, A052563, A084262.

a:=series(sqrt((1 - x)/(1 - 3*x)), x=0, 20): seq(n!*coeff(a, x, n), n=0..19); # Paolo P. Lava, Mar 26 2019

nmax = 19; CoefficientList[Series[Sqrt[(1 - x)/(1 - 3*x)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n - 1, k - 1] (2 k - 1)!! n!/k!, {k, 0, n}], {n, 0, 19}]
Join[{1}, Table[n! Hypergeometric2F1[3/2, 1 - n, 2, -2], {n, 19}]]

my(x='x + O('x^25)); Vec(serlaplace(sqrt((1 - x)/(1 - 3*x)))) \\ Michel Marcus, Mar 26 2019

a(n) = Sum_{k=0..n} binomial(n-1,k-1)*(2*k-1)!!*n!/k!.
a(n) ~ 2 * 3^(n - 1/2) * n^n / exp(n). - Vaclav Kotesovec, Mar 26 2019
D-finite with recurrence: (3*n^2 + 3*n)*a(n) + (-5 - 4*n)*a(n + 1) + a(n + 2)=0. - Robert Israel, Mar 26 2019

A318223 Expansion of e.g.f. exp(x/(1 + 2*x)).

Original entry on oeis.org

1, 1, -3, 13, -71, 441, -2699, 9157, 206193, -8443151, 236126701, -6169406979, 161388751657, -4327824442967, 120012465557349, -3450029411174219, 102741264191105761, -3160671409312412703, 99982488984008583133, -3230094912866216253971, 105481073534842477321881, -3423260541695907002392679

Offset: 0

Ilya Gutkovskiy, Aug 21 2018

sign

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A002866, A025168, A111884, A122803.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x/(1+2*x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Feb 07 2019

seq(n!*coeff(series(exp(x/(1+2*x)),x=0,22),x,n),n=0..21); # Paolo P. Lava, Jan 09 2019

nmax = 21; CoefficientList[Series[Exp[x/(1 + 2 x)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-2)^(n - k) Binomial[n - 1, k - 1] n!/k!, {k, 0, n}], {n, 0, 21}]
a[n_] := a[n] = Sum[(-2)^(k - 1) k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]
Join[{1}, Table[(-2)^(n - 1) n! Hypergeometric1F1[1 - n, 2, 1/2], {n, 21}]]

my(x='x+O('x^30)); Vec(serlaplace(exp(x/(1+2*x)))) \\ G. C. Greubel, Feb 07 2019

m = 30; T = taylor(exp(x/(1+2*x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Feb 07 2019

E.g.f.: Product_{k>=1} exp((-2)^(k-1)*x^k).
a(n) = Sum_{k=0..n} (-2)^(n-k)*binomial(n-1,k-1)*n!/k!.
a(0) = 1; a(n) = Sum_{k=1..n} (-2)^(k-1)*k!*binomial(n-1,k-1)*a(n-k).

A322218 E.g.f.: C(x,q) = 1 + Integral S(x,q) * C(qx,q) dx, such that C(x,q)^2 - S(x,q)^2 = 1, where C(x,q) = Sum_{n>=0} sum_{k=0..n(n-1)/2} T(n,k)x^ny^k/n!, as an irregular triangle of coefficients T(n,k) read by rows.

Original entry on oeis.org

1, 1, 1, 4, 1, 20, 16, 24, 1, 56, 336, 288, 384, 128, 192, 1, 120, 2352, 6448, 12736, 5888, 10176, 5760, 3840, 1280, 1920, 1, 220, 10032, 93280, 214016, 472704, 385472, 431616, 294912, 341504, 141056, 164352, 69120, 46080, 15360, 23040, 1, 364, 32032, 740168, 4072640, 11702912, 18676672, 30112640, 23848704, 27599616, 17884032, 20958208, 13595136, 11074560, 5992448, 5945856, 2673664, 2300928, 967680, 645120, 215040, 322560, 1, 560, 84448, 3952832, 53301248, 230161152, 738249344, 1166436352, 1970874368, 2196244480, 2459786240, 1804101632, 2061498368, 1537437696, 1437724672, 989968384, 921092096, 487923712, 499621888, 282034176, 211599360, 117383168, 108036096, 42778624, 36814848, 15482880, 10321920, 3440640, 5160960

Offset: 0

Paul D. Hanna, Dec 16 2018

Compare to Jacobi's elliptic function cn(x,k) = 1 - Integral sn(x,k)*dn(x,k) dx such that cn(x,k)^2 + sn(x,k)^2 = 1 and dn(x,k)^2 + k^2*sn(x,k)^2 = 1.
Right border equals A002866.
Row sums equal the secant numbers (A000364).
Last n terms in row n of this triangle and of triangle A322219 are equal for n>0.

			E.g.f. C(x,q) = Sum_{n>=0} sum_{k=0..n*(n-1)/2} T(n,k) * x^(2*n)*q^(2*k)/(2*n)! starts
C(x,q) = 1 + x^2/2! + (4*q^2 + 1)*x^4/4! + (24*q^6 + 16*q^4 + 20*q^2 + 1)*x^6/6! + (192*q^12 + 128*q^10 + 384*q^8 + 288*q^6 + 336*q^4 + 56*q^2 + 1)*x^8/8! + (1920*q^20 + 1280*q^18 + 3840*q^16 + 5760*q^14 + 10176*q^12 + 5888*q^10 + 12736*q^8 + 6448*q^6 + 2352*q^4 + 120*q^2 + 1)*x^10/10! + (23040*q^30 + 15360*q^28 + 46080*q^26 + 69120*q^24 + 164352*q^22 + 141056*q^20 + 341504*q^18 + 294912*q^16 + 431616*q^14 + 385472*q^12 + 472704*q^10 + 214016*q^8 + 93280*q^6 + 10032*q^4 + 220*q^2 + 1)*x^12/12! + ...
such that C(x,q) = cosh( Integral C(q*x,q) dx ).
This irregular triangle of coefficients T(n,k) of x^(2*n)*q^(2*k)/(2*n)! in C(x,q) begins:
1;
1;
1, 4;
1, 20, 16, 24;
1, 56, 336, 288, 384, 128, 192;
1, 120, 2352, 6448, 12736, 5888, 10176, 5760, 3840, 1280, 1920;
1, 220, 10032, 93280, 214016, 472704, 385472, 431616, 294912, 341504, 141056, 164352, 69120, 46080, 15360, 23040;
1, 364, 32032, 740168, 4072640, 11702912, 18676672, 30112640, 23848704, 27599616, 17884032, 20958208, 13595136, 11074560, 5992448, 5945856, 2673664, 2300928, 967680, 645120, 215040, 322560;
1, 560, 84448, 3952832, 53301248, 230161152, 738249344, 1166436352, 1970874368, 2196244480, 2459786240, 1804101632, 2061498368, 1537437696, 1437724672, 989968384, 921092096, 487923712, 499621888, 282034176, 211599360, 117383168, 108036096, 42778624, 36814848, 15482880, 10321920, 3440640, 5160960; ...
RELATED SERIES.
S(x,q) = x + (q^2 + 1)*x^3/3! + (4*q^6 + q^4 + 10*q^2 + 1)*x^5/5! + (24*q^12 + 16*q^10 + 20*q^8 + 85*q^6 + 91*q^4 + 35*q^2 + 1)*x^7/7! + (192*q^20 + 128*q^18 + 384*q^16 + 288*q^14 + 1200*q^12 + 632*q^10 + 2737*q^8 + 1324*q^6 + 966*q^4 + 84*q^2 + 1)*x^9/9! + (1920*q^30 + 1280*q^28 + 3840*q^26 + 5760*q^24 + 10176*q^22 + 16448*q^20 + 19776*q^18 + 27568*q^16 + 49872*q^14 + 69816*q^12 + 64329*q^10 + 50941*q^8 + 26818*q^6 + 5082*q^4 + 165*q^2 + 1)*x^11/11! +  ...
where C(x,q)^2 - S(x,q)^2 = 1.

Paul D. Hanna, Table of n, a(n) for n = 0..4525, as a flattened irregular triangle read by rows 0..30.

Cf. A322219 (S(x,q)), A000364 (row sums), A193544.

rows = 8; m = 2 rows; s[x_, ] = x; c[, ] = 1; Do[s[x, q_] = Integrate[c[x, q] c[q x, q] + O[x]^m // Normal, x]; c[x_, q_] = 1 + Integrate[s[x, q] c[q x, q] + O[x]^m // Normal, x], {m}];
CoefficientList[#, q^2]& /@ (CoefficientList[c[x, q], x] Range[0, m]!) // DeleteCases[#, {}]& // Flatten (* Jean-François Alcover, Dec 17 2018 *)

{T(n,k) = my(S=x,C=1); for(i=1,2*n,
S = intformal(C*subst(C,x,q*x) +O(x^(2*n+1)));
C = 1 + intformal(S*subst(C,x,q*x)));
(2*n)!*polcoeff( polcoeff(C,2*n,x),2*k,q)}
for(n=0,10, for(k=0,n*(n-1)/2, print1( T(n,k),", "));print(""))

E.g.f. C(x,q) and related series S(x,q) satisfy:
(1) C(x,q)^2 - S(x,q)^2 = 1.
(2) C(x,q) = 1 + Integral S(x,q) * C(q*x,q) dx.
(3) S(x,q) = Integral C(x,q) * C(q*x,q) dx.
(4a) C(x,q) + S(x,q) = exp( Integral C(q*x,q) dx ).
(4b) C(x,q) = cosh( Integral C(q*x,q) dx ).
(4c) S(x,q) = sinh( Integral C(q*x,q) dx ).
(5) C(q*x,q) = 1 + q * Integral S(q*x,q) * C(q^2*x,q) dx.
(6) S(q*x,q) = q * Integral C(q*x,q) * C(q^2*x,q) dx.
(7a) C(q*x,q) + S(q*x,q) = exp( q * Integral C(q^2*x,q) dx ).
(7b) C(q*x,q) = cosh( q * Integral C(q^2*x,q) dx ).
(7c) S(q*x,q) = sinh( q * Integral C(q^2*x,q) dx ).
PARTICULAR ARGUMENTS.
C(x,q=0) = cosh(x).
C(x,q=1) = 1/cos(x).
C(x,q=i) = cl(i*x), where cl(x) is the cosine lemniscate function (A159600).
FORMULAS FOR TERMS.
T(n, n*(n-1)/2) = 2^(n-1)*n! for n >= 1.
T(n, n*(n-1)/2 - k) = A322219(n, n*(n+1)/2 - k) for k = 0..n-1, n > 0.
Sum_{k=0..n*(n-1)/2} T(n,k) = A000364(n) for n >= 0.
Sum_{k=0..n*(n-1)/2} T(n,k)*(-1)^k = A193544(2*n+1) for n >= 0.

A322219 E.g.f.: S(x,q) = Integral C(x,q) * C(qx,q) dx, such that C(x,q)^2 - S(x,q)^2 = 1, where S(x,q) = Sum_{n>=0} sum_{k=0..n(n+1)/2} T(n,k)x^ny^k/n!, as an irregular triangle of coefficients T(n,k) read by rows.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 4, 1, 35, 91, 85, 20, 16, 24, 1, 84, 966, 1324, 2737, 632, 1200, 288, 384, 128, 192, 1, 165, 5082, 26818, 50941, 64329, 69816, 49872, 27568, 19776, 16448, 10176, 5760, 3840, 1280, 1920, 1, 286, 18447, 279136, 954239, 2550054, 2455233, 4013788, 2929104, 3264864, 1176640, 1815552, 834752, 731136, 394752, 491264, 141056, 164352, 69120, 46080, 15360, 23040, 1, 455, 53053, 1780207, 15627183, 51699869, 128611679, 187372653, 213804652, 257006976, 245800968, 195109120, 161177792, 123750592, 83792000, 69316224, 52893696, 35140992, 28215808, 18433536, 12687360, 8411648, 5945856, 2673664, 2300928, 967680, 645120, 215040, 322560

Offset: 0

Paul D. Hanna, Dec 16 2018

Compare to Jacobi's elliptic function sn(x,k) = Integral cn(x,k)*dn(x,k) dx such that cn(x,k)^2 + sn(x,k)^2 = 1 and dn(x,k)^2 + k^2*sn(x,k)^2 = 1.
Right border equals A002866.
Row sums equal the tangent numbers (A000182).
Last n terms in row n of this triangle and of triangle A322218 are equal for n>0.

			E.g.f. S(x,q) = Sum_{n>=0} sum_{k=0..n*(n+1)/2} T(n,k)*x^(2*n+1)*q^(2*k)/(2*n+1)! starts
S(x,q) = x + (q^2 + 1)*x^3/3! + (4*q^6 + q^4 + 10*q^2 + 1)*x^5/5! + (24*q^12 + 16*q^10 + 20*q^8 + 85*q^6 + 91*q^4 + 35*q^2 + 1)*x^7/7! + (192*q^20 + 128*q^18 + 384*q^16 + 288*q^14 + 1200*q^12 + 632*q^10 + 2737*q^8 + 1324*q^6 + 966*q^4 + 84*q^2 + 1)*x^9/9! + (1920*q^30 + 1280*q^28 + 3840*q^26 + 5760*q^24 + 10176*q^22 + 16448*q^20 + 19776*q^18 + 27568*q^16 + 49872*q^14 + 69816*q^12 + 64329*q^10 + 50941*q^8 + 26818*q^6 + 5082*q^4 + 165*q^2 + 1)*x^11/11! + ...
such that S(x,q) = sinh( Integral C(q*x,q) dx ) and C(x,q)^2 = 1 + S(x,q)^2.
This irregular triangle of coefficients T(n,k) of x^(2*n+1)*q^(2*k)/(2*n+1)! in S(x,q) begins:
1;
1, 1;
1, 10, 1, 4;
1, 35, 91, 85, 20, 16, 24;
1, 84, 966, 1324, 2737, 632, 1200, 288, 384, 128, 192;
1, 165, 5082, 26818, 50941, 64329, 69816, 49872, 27568, 19776, 16448, 10176, 5760, 3840, 1280, 1920;
1, 286, 18447, 279136, 954239, 2550054, 2455233, 4013788, 2929104, 3264864, 1176640, 1815552, 834752, 731136, 394752, 491264, 141056, 164352, 69120, 46080, 15360, 23040;
1, 455, 53053, 1780207, 15627183, 51699869, 128611679, 187372653, 213804652, 257006976, 245800968, 195109120, 161177792, 123750592, 83792000, 69316224, 52893696, 35140992, 28215808, 18433536, 12687360, 8411648, 5945856, 2673664, 2300928, 967680, 645120, 215040, 322560;
1, 680, 129948, 8212360, 163115238, 1001312104, 3705217660, 7815443320, 15434182497, 17298854576, 23429393056, 21144463040, 25624143104, 18454639872, 18756800128, 12036914176, 12076688384, 7122865152, 7609525248, 4420732928, 4042876928, 2553473024, 2465701888, 1353586688, 1234018304, 619528192, 587358208, 311279616, 255467520, 117383168, 108036096, 42778624, 36814848, 15482880, 10321920, 3440640, 5160960; ...
RELATED SERIES.
C(x,q) = 1 + x^2/2! + (4*q^2 + 1)*x^4/4! + (24*q^6 + 16*q^4 + 20*q^2 + 1)*x^6/6! + (192*q^12 + 128*q^10 + 384*q^8 + 288*q^6 + 336*q^4 + 56*q^2 + 1)*x^8/8! + (1920*q^20 + 1280*q^18 + 3840*q^16 + 5760*q^14 + 10176*q^12 + 5888*q^10 + 12736*q^8 + 6448*q^6 + 2352*q^4 + 120*q^2 + 1)*x^10/10! + (23040*q^30 + 15360*q^28 + 46080*q^26 + 69120*q^24 + 164352*q^22 + 141056*q^20 + 341504*q^18 + 294912*q^16 + 431616*q^14 + 385472*q^12 + 472704*q^10 + 214016*q^8 + 93280*q^6 + 10032*q^4 + 220*q^2 + 1)*x^12/12! + ...
such that C(x,q) = cosh( Integral C(q*x,q) dx ).

Paul D. Hanna, Table of n, a(n) for n = 0..4990, as a flattened irregular triangle read by rows 0..30.

Cf. A322218 (C(x,q)), A000182 (row sums), A104203, A002866.

rows = 8; m = 2 rows; s[x_, ] = x; c[, ] = 1; Do[s[x, q_] = Integrate[c[x, q] c[q x, q] + O[x]^m // Normal, x]; c[x_, q_] = 1 + Integrate[s[x, q] c[q x, q] + O[x]^m // Normal, x], {m}];
CoefficientList[#, q^2]& /@ (CoefficientList[s[x, q], x] Range[0, m-1]!) // DeleteCases[#, {}]& // Flatten (* Jean-François Alcover, Dec 17 2018 *)

{T(n,k) = my(S=x,C=1); for(i=1,2*n,
S = intformal(C*subst(C,x,q*x) +O(x^(2*n+1)));
C = 1 + intformal(S*subst(C,x,q*x)));
(2*n+1)!*polcoeff( polcoeff(S,2*n+1,x),2*k,q)}
for(n=0,10, for(k=0,n*(n+1)/2, print1( T(n,k),", "));print(""))

E.g.f. S(x,q) and related series C(x,q) satisfy:
(1) C(x,q)^2 - S(x,q)^2 = 1.
(2) C(x,q) = 1 + Integral S(x,q) * C(q*x,q) dx.
(3) S(x,q) = Integral C(x,q) * C(q*x,q) dx.
(4a) C(x,q) + S(x,q) = exp( Integral C(q*x,q) dx ).
(4b) C(x,q) = cosh( Integral C(q*x,q) dx ).
(4c) S(x,q) = sinh( Integral C(q*x,q) dx ).
(5) C(q*x,q) = 1 + q * Integral S(q*x,q) * C(q^2*x,q) dx.
(6) S(q*x,q) = q * Integral C(q*x,q) * C(q^2*x,q) dx.
(7a) C(q*x,q) + S(q*x,q) = exp( q * Integral C(q^2*x,q) dx ).
(7b) C(q*x,q) = cosh( q * Integral C(q^2*x,q) dx ).
(7c) S(q*x,q) = sinh( q * Integral C(q^2*x,q) dx ).
PARTICULAR ARGUMENTS.
S(x,q=0) = sinh(x).
S(x,q=1) = tan(x).
S(x,q=i) = -i * sl(i*x), where sl(x) is the sine lemniscate function (A104203).
FORMULAS FOR TERMS.
T(n, n*(n+1)/2) = 2^(n-1)*n! for n >= 1.
T(n, n*(n+1)/2 - k) = A322218(n, n*(n-1)/2 - k) for k = 0..n-1, n > 0.
Sum_{k=0..n*(n+1)/2} T(n,k) = A000182(n+1) for n >= 0.
Sum_{k=0..n*(n+1)/2} T(n,k)*(-1)^k = A104203(2*n+1) for n >= 0.

A332257 E.g.f.: (1 - sinh(x)) / (1 - 2*sinh(x)).

Original entry on oeis.org

1, 1, 4, 25, 208, 2161, 26944, 391945, 6515968, 121866721, 2532496384, 57890223865, 1443611004928, 38999338931281, 1134616226381824, 35367467110007785, 1175946733416153088, 41543231955279099841, 1553948045857778827264, 61355543097139813855705

Offset: 0

Ilya Gutkovskiy, Feb 08 2020

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..401
Paul Kinlaw, Michael Morris, and Samanthak Thiagarajan, Sums related to the Fibonacci sequence, Husson University (2021).

Cf. A000557, A000670, A002866, A006154, A050351, A331608.

nmax = 19; CoefficientList[Series[(1 - Sinh[x])/(1 - 2 Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!

seq(n)={Vec(serlaplace((1 - sinh(x + O(x*x^n))) / (1 - 2*sinh(x + O(x*x^n)))))} \\ Andrew Howroyd, Feb 08 2020

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A006154(k) * a(n-k).

a(n) ~ n! / (2*sqrt(5) * log((1 + sqrt(5))/2)^(n+1)). - Vaclav Kotesovec, Feb 08 2020

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A322218 E.g.f.: C(x,q) = 1 + Integral S(x,q) * C(q*x,q) dx, such that C(x,q)^2 - S(x,q)^2 = 1, where C(x,q) = Sum_{n>=0} sum_{k=0..n*(n-1)/2} T(n,k)*x^n*y^k/n!, as an irregular triangle of coefficients T(n,k) read by rows.

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A322218 E.g.f.: C(x,q) = 1 + Integral S(x,q) * C(qx,q) dx, such that C(x,q)^2 - S(x,q)^2 = 1, where C(x,q) = Sum_{n>=0} sum_{k=0..n(n-1)/2} T(n,k)x^ny^k/n!, as an irregular triangle of coefficients T(n,k) read by rows.

A322219 E.g.f.: S(x,q) = Integral C(x,q) * C(qx,q) dx, such that C(x,q)^2 - S(x,q)^2 = 1, where S(x,q) = Sum_{n>=0} sum_{k=0..n(n+1)/2} T(n,k)x^ny^k/n!, as an irregular triangle of coefficients T(n,k) read by rows.