0,3

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))).

Tanya Khovanova, Recursive Sequences

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))).

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^n-1 for n>0:

6^1-1 = 1+4 = 5

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

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

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

6^5-1 = 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((1-x)/(1-5*x-x^2)).

(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)))}

Cf. A100234, A100236, A100237, A100232.

Adjacent sequences: A100232 A100233 A100234 this_sequence A100236 A100237 A100238

Sequence in context: A125129 A013611 A077910 this_sequence A089072 A036177 A143469

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Nov 29 2004