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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!.

%I A121408 #38 Aug 01 2025 19:04:03
%S A121408 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,
%T A121408 2304,0,784,0,56,0,1,11025,0,12916,0,1974,0,84,0,1,0,147456,0,52480,0,
%U A121408 4368,0,120,0,1,893025,0,1057221,0,172810,0,8778,0,165,0,1,0,14745600,0
%N 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!.
%C A121408 Row sums are equal to A006228(n). This is sequence A091885 with additional intertwining zeros.
%C A121408 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
%C A121408 The unsigned matrix inverse is A136630 (with a different offset). - _Peter Bala_, Feb 23 2011
%C A121408 Also the Bell transform of A177145. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 27 2016
%D A121408 B. C. Berndt, Ramanujan's Notebooks Part 1, Springer-Verlag 1985.
%H A121408 Vladimir Kruchinin, <a href="https://arxiv.org/abs/1009.2565"> Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.
%F A121408 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]
%F A121408 From _Peter Bala_, Aug 29 2012: (Start)
%F A121408 See A182971 for a version of the row reverse of this triangle.
%F A121408 Even-indexed row polynomial R(2*n,x) = x^2*prod(k=1..n-1, (x^2 + (2*k)^2) ).
%F A121408 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)
%F A121408 Sum_{k=0..n} T(n+1,k+1)*A000111(k) = n! = A000142(n). - _Alexander Burstein_, Aug 01 2025
%e A121408 Triangle starts:
%e A121408   1;
%e A121408   0,1;
%e A121408   1,0,1;
%e A121408   0,4,0,1;
%e A121408   9,0,10,0,1;
%e A121408   0,64,0,20,0,1;
%e A121408 Row polynomials R(6,x) = x^2*(x^2 + 2^2)*(x^2 + 4^2) = 64*x^2 + 20*x^4 + x^6 and
%e A121408 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
%p A121408 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
%p A121408 # The function BellMatrix is defined in A264428.
%p A121408 # Adds (1,0,0,0, ..) as column 0.
%p A121408 BellMatrix(n -> `if`(n::odd,0,doublefactorial(n-1)^2), 9); # _Peter Luschny_, Jan 27 2016
%t A121408 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A121408 rows = 12;
%t A121408 M = BellMatrix[If[OddQ[#], 0, (# - 1)!!^2] &, rows];
%t A121408 Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 24 2018, after _Peter Luschny_ *)
%Y A121408 Cf. A000111, A000142, A006228, A091885, A136630. A182971.
%K A121408 nonn,tabl
%O A121408 1,8
%A A121408 _Emeric Deutsch_, Jul 28 2006