The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60, we have over 367,000 sequences, and we’ve crossed 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A100235 Triangle, read by rows, of the coefficients of [x^k] in G100234(x)^n such that the row sums are 6^n-1 for n>0, where G100234(x) is the g.f. of A100234. 3

%I #14 Jun 13 2017 22:14:23

%S 1,1,4,1,8,26,1,12,63,139,1,16,116,436,726,1,20,185,965,2830,3774,1,

%T 24,270,1790,7335,17634,19601,1,28,371,2975,15505,52444,106827,101784,

%U 1,32,488,4584,28860,124424,358748,633952,528526,1,36,621,6681,49176,256194

%N Triangle, read by rows, of the coefficients of [x^k] in G100234(x)^n such that the row sums are 6^n-1 for n>0, where G100234(x) is the g.f. of A100234.

%C 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^n-b^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+(m-1)*x+sqrt(1+2*(m-2*b-1)*x+(m^2-2*m+4*b+1)*x^2))/2; the triangle formed from powers of F(x) will have the g.f.: G(x,y)=(1-2*x*y+m*x^2*y^2)/((1-x*y)*(1-(m-1)*x*y-x^2*y^2-x*(1-x*y))).

%F G.f.: A(x, y)=(1-2*x*y+6*x^2*y^2)/((1-x*y)*(1-5*x*y-x^2*y^2-x*(1-x*y))).

%e Rows begin:

%e [1],

%e [1,4],

%e [1,8,26],

%e [1,12,63,139],

%e [1,16,116,436,726],

%e [1,20,185,965,2830,3774],

%e [1,24,270,1790,7335,17634,19601],

%e [1,28,371,2975,15505,52444,106827,101784],

%e [1,32,488,4584,28860,124424,358748,633952,528526],...

%e where row sums form 6^n-1 for n>0:

%e 6^1-1 = 1+4 = 5

%e 6^2-1 = 1+8+26 = 35

%e 6^3-1 = 1+12+63+139 = 215

%e 6^4-1 = 1+16+116+436+726 = 1295

%e 6^5-1 = 1+20+185+965+2830+3774 = 7775.

%e The main diagonal forms A100236 = [1,4,26,139,726,3774,...],

%e where Sum_{n>=1} A100236(n)/n*x^n = log((1-x)/(1-5*x-x^2)).

%t 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}]](* _Jean-François Alcover_, May 11 2012, after PARI *)

%o (PARI) T(n,k,m=6)=if(n<k || k<0,0,if(k==0,1, polcoeff(((1+(m-1)*x+sqrt(1+2*(m-3)*x+(m^2-2*m+5)*x^2+x*O(x^k)))/2)^n,k)))

%Y Cf. A100234, A100236, A100237, A100232.

%K nonn,tabl

%O 0,3

%A _Paul D. Hanna_, Nov 29 2004

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 2 11:04 EST 2023. Contains 367517 sequences. (Running on oeis4.)