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 A121408 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). 4
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 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..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 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 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. 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). [From 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) EXAMPLE 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 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 *) CROSSREFS Cf. A006228, A091885, A136630. A182971. Sequence in context: A079642 A342911 A221483 * A186761 A199786 A189245 Adjacent sequences:  A121405 A121406 A121407 * A121409 A121410 A121411 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Jul 28 2006 STATUS approved

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Last modified September 18 09:05 EDT 2021. Contains 347518 sequences. (Running on oeis4.)