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A100229 Triangle, read by rows, of the coefficients of [x^k] in G100228(x)^n such that the row sums are 4^n-1 for n>0, where G100228(x) is the g.f. of A100228. 3
1, 1, 2, 1, 4, 10, 1, 6, 21, 35, 1, 8, 36, 92, 118, 1, 10, 55, 185, 380, 392, 1, 12, 78, 322, 879, 1506, 1297, 1, 14, 105, 511, 1715, 3948, 5803, 4286, 1, 16, 136, 760, 3004, 8536, 17020, 21904, 14158, 1, 18, 171, 1077, 4878, 16344, 40395, 71109, 81387, 46763 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The main diagonal forms A100230. Secondary diagonal is T(n+1,n) = (n+1)*A052924(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))).

LINKS

Table of n, a(n) for n=0..54.

FORMULA

G.f.: A(x, y)=(1-2*x*y+4*x^2*y^2)/((1-x*y)*(1-3*x*y-x^2*y^2-x*(1-x*y))).

EXAMPLE

Rows begin:

[1],

[1,2],

[1,4,10],

[1,6,21,35],

[1,8,36,92,118],

[1,10,55,185,380,392],

[1,12,78,322,879,1506,1297],

[1,14,105,511,1715,3948,5803,4286],

[1,16,136,760,3004,8536,17020,21904,14158],...

where row sums form 4^n-1 for n>0:

4^1-1 = 1+2 = 3

4^2-1 = 1+4+10 = 15

4^3-1 = 1+6+21+35 = 63

4^4-1 = 1+8+36+92+118 = 255

4^5-1 = 1+10+55+185+380+392 = 1023.

The main diagonal forms A100230 = [1,2,10,35,118,392,1297,...],

where Sum_{n>=1} A100230(n)/n*x^n = log((1-x)/(1-3*x-x^2)).

PROG

(PARI) T(n, k, m=4)=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)))

CROSSREFS

Cf. A100228, A100230, A052924.

Sequence in context: A117338 A279927 A137634 * A071949 A297506 A297720

Adjacent sequences:  A100226 A100227 A100228 * A100230 A100231 A100232

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Nov 29 2004

STATUS

approved

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Last modified June 21 00:57 EDT 2021. Contains 345329 sequences. (Running on oeis4.)