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A126265 Triangle of coefficients of q in e.g.f. that satisfies: A(x,q) = exp( q*x*A(q*x,q) ), read by rows of [n*(n-1)/2 + 1] terms in row n for n>=0. 7
1, 1, 1, 2, 1, 6, 3, 6, 1, 12, 24, 28, 24, 12, 24, 1, 20, 90, 140, 245, 120, 240, 140, 120, 60, 120, 1, 30, 240, 660, 1320, 1626, 1920, 2100, 1560, 1830, 1440, 1440, 840, 720, 360, 720, 1, 42, 525, 2450, 6195, 12432, 15127, 23310, 21000, 26250, 19320, 26502, 19320 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row sums equals A000272(n) = (n+1)^(n-1). Last term in rows are the factorials. Coefficients of q in {[x^n] A(x,q)} when read backward converge to the sequence: [1,1/2,1,7/6,2,2,85/24,11/3,65/12,19/3,357/40,19/2,111/8,123/8,81/4,...].

LINKS

Paul D. Hanna, Rows n=0..18, flattened.

FORMULA

G.f.: A(x,q) = Sum_{n>=0} (x^n/n!)*q^n*Sum_{k=0..n*(n-1)/2} T(n,k)*q^k.

EXAMPLE

G.f.: A(x,q) = 1 + q*x + (1 + 2*q)*q^2*x^2/2! +

(1 + 6*q + 3*q^2 + 6*q^3)*q^3*x^3/3! +

(1 + 12*q + 24*q^2 + 28*q^3 + 24*q^4 + 12*q^5 + 24*q^6)*q^4*x^4/4! +...

Triangle begins:

1;

1;

1, 2;

1, 6, 3, 6;

1, 12, 24, 28, 24, 12, 24;

1, 20, 90, 140, 245, 120, 240, 140, 120, 60, 120;

1, 30, 240, 660, 1320, 1626, 1920, 2100, 1560, 1830, 1440, 1440, 840, 720, 360, 720; ...

PROG

(PARI) {T(n, k)=local(A=x); for(i=1, n, A=x*exp(subst(A, x, q*x+x*O(x^n)))); if(k>n*(n-1)/2|k<0, 0, Vec(Vec(A)[n+1]*n!+q*O(q^(n*(n+1)/2)))[k+1])}

for(n=0, 9, for(k=0, n*(n-1)/2, print1(T(n, k), ", ")); print(""))

CROSSREFS

Cf. A000272 (row sums); diagonals: A126266, A126267.

Sequence in context: A171178 A100014 A062566 * A124441 A026191 A050137

Adjacent sequences:  A126262 A126263 A126264 * A126266 A126267 A126268

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Dec 22 2006

STATUS

approved

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Last modified October 22 18:03 EDT 2014. Contains 248400 sequences.