A121408 Triangle T(n,k) defined by the generating function: exp(y*arcsin(x))-1 = Sum_{n>=1} (Sum_{k=1..n} T(n,k)*y^k)*x^n/n!.

1, 0, 1, 1, 0, 1, 0, 4, 0, 1, 9, 0, 10, 0, 1, 0, 64, 0, 20, 0, 1, 225, 0, 259, 0, 35, 0, 1, 0, 2304, 0, 784, 0, 56, 0, 1, 11025, 0, 12916, 0, 1974, 0, 84, 0, 1, 0, 147456, 0, 52480, 0, 4368, 0, 120, 0, 1, 893025, 0, 1057221, 0, 172810, 0, 8778, 0, 165, 0, 1, 0, 14745600, 0

Offset: 1

Author

Emeric Deutsch, Jul 28 2006

Keywords

• nonn
• tabl

Comments

Row sums are equal to A006228(n). This is sequence A091885 with additional intertwining zeros.

F(n,m) = n!*T(n,m)/m! is a composite (akin to Riordan arrays) of F(x)=arcsin(x) and (F(x))^m = Sum_{n>=m} F(n,m)*x^n, and for o.g.f. G(x), G(arcsin(x)) = g(0) +Sum_{n>=1} Sum_{m=1..n} F(n,m)*g(m)*x^n, see the preprint. - Vladimir Kruchinin, Feb 10 2011

The unsigned matrix inverse is A136630 (with a different offset). - Peter Bala, Feb 23 2011

Also the Bell transform of A177145. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

Examples

			Triangle starts:
  1;
  0,1;
  1,0,1;
  0,4,0,1;
  9,0,10,0,1;
  0,64,0,20,0,1;
Row polynomials R(6,x) = x^2*(x^2 + 2^2)*(x^2 + 4^2) = 64*x^2 + 20*x^4 + x^6 and
R(7,x) = x*(x^2 + 1)*(x^2 + 3^2)*(x^2 + 5^2) = 225*x + 259*x^3 + 35*x^5 + x^7. - _Peter Bala_, Aug 29 2012
		

References

• B. C. Berndt, Ramanujan's Notebooks Part 1, Springer-Verlag 1985.

Links

• Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Crossrefs

Cf. A000111, A000142, A006228, A091885, A136630. A182971.

Programs

Maple

g:=exp(y*arcsin(x))-1: gser:=simplify(series(g,x=0,15)): for n from 1 to 12 do P[n]:=sort(n!*coeff(gser,x,n)) od: for n from 1 to 12 do seq(coeff(P[n],y,k),k=1..n) od; # yields sequence in triangular form
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n::odd,0,doublefactorial(n-1)^2), 9); # Peter Luschny, Jan 27 2016

Mathematica

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[OddQ[#], 0, (# - 1)!!^2] &, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

Formula

T(n,m) = ((n-1)!/(m-1)!) *sum_{k=1..n-m} sum_{j=1..k} binomial(k,j) *(2^(1-j) /(n-m+j)!) *sum{i=0..floor(j/2)} (-1)^((n-m)/2-i-j) *binomial(j,i) *(j-2*i)^(n-m+j) *binomial(k+n-1,n-1), n>m and even(n-m). [Vladimir Kruchinin, Feb 10 2011]

From Peter Bala, Aug 29 2012: (Start)

See A182971 for a version of the row reverse of this triangle.

Even-indexed row polynomial R(2*n,x) = x^2*prod(k=1..n-1, (x^2 + (2*k)^2) ).

Odd-indexed row polynomial R(2*n+1,x) = x*prod(k=1..n, (x^2 + (2*k-1)^2) ). See Berndt p.263. (End)

Sum_{k=0..n} T(n+1,k+1)*A000111(k) = n! = A000142(n). - Alexander Burstein, Aug 01 2025