A121408 - OEIS

%I #25 Jun 24 2018 06:37:58

%S 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 2304,0,784,0,56,0,1,11025,0,12916,0,1974,0,84,0,1,0,147456,0,52480,0,

%U 4368,0,120,0,1,893025,0,1057221,0,172810,0,8778,0,165,0,1,0,14745600,0

%N Triangle T(n,k) defined by the generating function (in Maple notation): exp(y*arcsin(x))-1 = sum( sum(T(n,k)*y^k, k=1..n)*x^n/n!, n=1..infinity).

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

%C 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..infinity} F(n,m)*x^n, and for o.g.f. G(x), G(arcsin(x)) = g(0) +sum_{n=1..infinity} sum_{m=1..n} F(n,m)*g(m)*x^n, see the preprint. - Vladimir Kruchinin, Feb 10 2011

%C The unsigned matrix inverse is A136630  (with a different offset) - Peter

%C Bala, Feb 23 2011.

%C Also the Bell transform of A177145. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 27 2016

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

%H Vladimir Kruchinin, <a href="https://arxiv.org/abs/1009.2565"> Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.

%F 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). [From Vladimir Kruchinin, Feb 10 2011]

%F From _Peter Bala_, Aug 29 2012: (Start)

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

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

%F 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)

%e Triangle starts:

%e   1;

%e   0,1;

%e   1,0,1;

%e   0,4,0,1;

%e   9,0,10,0,1;

%e   0,64,0,20,0,1;

%e 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 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 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 # The function BellMatrix is defined in A264428.

%p # Adds (1,0,0,0, ..) as column 0.

%p BellMatrix(n -> `if`(n::odd,0,doublefactorial(n-1)^2), 9); # _Peter Luschny_, Jan 27 2016

%t BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

%t rows = 12;

%t M = BellMatrix[If[OddQ[#], 0, (# - 1)!!^2] &, rows];

%t Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 24 2018, after _Peter Luschny_ *)

%Y Cf. A006228, A091885, A136630. A182971.

%K nonn,tabl

%O 1,8

%A _Emeric Deutsch_, Jul 28 2006