

A100235


Triangle, read by rows, of the coefficients of [x^k] in G100234(x)^n such that the row sums are 6^n1 for n>0, where G100234(x) is the g.f. of A100234.


3



1, 1, 4, 1, 8, 26, 1, 12, 63, 139, 1, 16, 116, 436, 726, 1, 20, 185, 965, 2830, 3774, 1, 24, 270, 1790, 7335, 17634, 19601, 1, 28, 371, 2975, 15505, 52444, 106827, 101784, 1, 32, 488, 4584, 28860, 124424, 358748, 633952, 528526, 1, 36, 621, 6681, 49176, 256194
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OFFSET

0,3


COMMENTS

The main diagonal forms A100236. Secondary diagonal is: T(n+1,n) = (n+1)*A100237(n). More generally, if g.f. F(x) satisfies: m^nb^n = Sum_{k=0..n} [x^k]F(x)^n, then F(x) also satisfies: (m+z)^n  (b+z)^n + z^n = Sum_{k=0..n} [x^k](F(x)+z*x)^n for all z and F(x)=(1+(m1)*x+sqrt(1+2*(m2*b1)*x+(m^22*m+4*b+1)*x^2))/2; the triangle formed from powers of F(x) will have the g.f.: G(x,y)=(12*x*y+m*x^2*y^2)/((1x*y)*(1(m1)*x*yx^2*y^2x*(1x*y))).


LINKS

Table of n, a(n) for n=0..50.
Tanya Khovanova, Recursive Sequences


FORMULA

G.f.: A(x, y)=(12*x*y+6*x^2*y^2)/((1x*y)*(15*x*yx^2*y^2x*(1x*y))).


EXAMPLE

Rows begin:
[1],
[1,4],
[1,8,26],
[1,12,63,139],
[1,16,116,436,726],
[1,20,185,965,2830,3774],
[1,24,270,1790,7335,17634,19601],
[1,28,371,2975,15505,52444,106827,101784],
[1,32,488,4584,28860,124424,358748,633952,528526],...
where row sums form 6^n1 for n>0:
6^11 = 1+4 = 5
6^21 = 1+8+26 = 35
6^31 = 1+12+63+139 = 215
6^41 = 1+16+116+436+726 = 1295
6^51 = 1+20+185+965+2830+3774 = 7775.
The main diagonal forms A100236 = [1,4,26,139,726,3774,...],
where Sum_{n>=1} A100236(n)/n*x^n = log((1x)/(15*xx^2)).


MATHEMATICA

row[n_] := CoefficientList[ Series[ (1 + 5*x + Sqrt[1 + 6*x + 29*x^2])^n/2^n, {x, 0, n}], x]; Flatten[ Table[ row[n], {n, 0, 9}]](* JeanFrançois Alcover, May 11 2012, after PARI *)


PROG

(PARI) {T(n, k, m=6)=if(n<kk<0, 0, if(k==0, 1, polcoeff(((1+(m1)*x+sqrt(1+2*(m3)*x+(m^22*m+5)*x^2+x*O(x^k)))/2)^n, k)))}


CROSSREFS

Cf. A100234, A100236, A100237, A100232.
Sequence in context: A125129 A013611 A077910 * A089072 A036177 A177841
Adjacent sequences: A100232 A100233 A100234 * A100236 A100237 A100238


KEYWORD

nonn,tabl


AUTHOR

Paul D. Hanna, Nov 29 2004


STATUS

approved



