A249564 -id:A249564 - cp's OEIS frontend

A177385 E.g.f.: Sum_{n>=0} Product_{k=1..n} sinh(k*x).

1, 1, 4, 37, 616, 16081, 605164, 31011457, 2076192976, 175951716481, 18411425885524, 2331339303739777, 351341718484191736, 62144180030978834881, 12748469150999320273084, 3002313213700366145858497

Offset: 0

Paul D. Hanna, May 15 2010

nonn

Compare to the e.g.f. for A002105, the reduced tangent numbers:
. Sum_{n>=0} A002105(n+1)*x^n/n! = Sum_{n>=0} Product_{k=1..n} tanh(k*x).
Limit n->infinity n!*A177386(n) / (2^n*A177385(n)) = 1. - Vaclav Kotesovec, Nov 06 2014

			E.g.f: A(x) = 1 + x + 4*x^2/2! + 37*x^3/3! + 616*x^4/4! +...
A(x) = 1 + sinh(x) + sinh(x)*sinh(2x) + sinh(x)*sinh(2x)*sinh(3x) + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Cf. A249698, A177386, A249564, A249748.

Table[n!*SeriesCoefficient[Sum[Product[Sinh[k*x],{k,1,j}],{j,0,n}],{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 01 2014 *)
nn=20; tab = ConstantArray[0,nn]; tab[[1]] = Series[Sinh[x],{x,0,nn}]; Do[tab[[k]] = Series[tab[[k-1]]*Sinh[k*x],{x,0,nn}],{k,2,nn}]; Flatten[{1,Rest[CoefficientList[Sum[tab[[k]],{k,1,nn}],x] * Range[0,nn]!]}] (* Vaclav Kotesovec, Nov 04 2014 (more efficient) *)

{a(n)=local(X=x+x*O(x^n),Egf);Egf=sum(m=0,n,prod(k=1,m,sinh(k*X)));n!*polcoeff(Egf,n)}

a(n) ~ c * d^n * (n!)^2, where d = A249748 = 1.04689919262595424111342518325311817976789046475647184115584744582777576864..., c = 0.880333778211172907563073031129920597506533414605109200048966773434616066... . - Vaclav Kotesovec, Nov 04 2014

A334781 Array read by antidiagonals: T(n,k) = Sum_{i=1..n} binomial(1+i,2)^k.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 10, 10, 4, 0, 1, 28, 46, 20, 5, 0, 1, 82, 244, 146, 35, 6, 0, 1, 244, 1378, 1244, 371, 56, 7, 0, 1, 730, 8020, 11378, 4619, 812, 84, 8, 0, 1, 2188, 47386, 108020, 62003, 13880, 1596, 120, 9, 0, 1, 6562, 282124, 1047386, 867395, 256484, 35832, 2892, 165, 10

Offset: 0

Andrew Howroyd, May 15 2020

			Array begins:
===============================================================
n\k | 0  1    2     3      4        5         6           7
----|----------------------------------------------------------
  0 | 0  0    0     0      0        0         0           0 ...
  1 | 1  1    1     1      1        1         1           1 ...
  2 | 2  4   10    28     82      244       730        2188 ...
  3 | 3 10   46   244   1378     8020     47386      282124 ...
  4 | 4 20  146  1244  11378   108020   1047386    10282124 ...
  5 | 5 35  371  4619  62003   867395  12438011   181141499 ...
  6 | 6 56  812 13880 256484  4951496  98204132  1982230040 ...
  7 | 7 84 1596 35832 871140 22161864 580094436 15475158552 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Columns k=0..7 are A001477, A000292, A024166, A085438, A085439, A085440, A085441, A085442.
Rows n=0..3 are A000004, A000012, A034472, A074508.
Main diagonal is A249564(n > 0).
Cf. A154283 (coefficients).

T(n,k) = {sum(i=1, n, binomial(1+i,2)^k)}

T(n,k) = Sum_{i=0..2*(k-1)} A154283(k,i) * binomial(n+2+i, 2*k+i) for k > 0.

cp's OEIS Frontend

A177385 E.g.f.: Sum_{n>=0} Product_{k=1..n} sinh(k*x).

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