A183166 - OEIS

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A183166

G.f.: Sum_{n>=0} [Sum_{k=0..n} C(n,k)^2*x^k]^n*x^n.

1, 1, 2, 9, 46, 343, 3025, 32811, 417348, 6106921, 102307571, 1918139824, 40190540565, 928661958828, 23551552524966, 651213150740841, 19523328447786346, 631923020784069573, 21984209405892842663 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,3`
	LINKS	`Table of n, a(n) for n=0..18.`
	FORMULA	`G.f.: Sum_{n>=0} x^n(1-x)^(2n^2+n)[Sum_{k=0..n-1} C(n+k,k)^2*x^k]^n.`
	EXAMPLE	`G.f.: A(x) = 1 + x + 2x^2 + 9x^3 + 46x^4 + 343x^5 + 3025x^6 +...` `equals the sum of the series:` `A(x) = 1 + (1+x)x + (1 + 2^2x + x^2)^2x^2 +` `+ (1 + 3^2x + 3^2x^2 + x^3)^3x^3` `+ (1 + 4^2x + 6^2x^2 + 4^2x^3 + x^4)^4x^4` `+ (1 + 5^2x + 10^2x^2 + 10^2x^3 + 5^2x^4 + x^5)^5x^5` `+ (1 + 6^2x + 15^2x^2 + 20^2x^3 + 15^2x^4 + 6^2x^5 + x^6)^6x^6 +...` `The g.f. can also be expressed as:` `A(x) = 1 + x(1-x)^3(1 + 2^2x + 3^2x^2 + 4^2x^3 + 5^2x^4 +...)` `+ x^2(1-x)^10(1 + 3^2x + 6^2x^2 + 10^2x^3 + 15^2x^4 +...)^2` `+ x^3(1-x)^21(1 + 4^2x + 10^2x^2 + 20^2x^3 + 35^2x^4 +...)^3` `+ x^4(1-x)^36(1 + 5^2x + 15^2x^2 + 35^2x^3 + 70^2x^4 +...)^4` `+ x^5(1-x)^55(1 + 6^2x + 21^2x^2 + 56^2x^3 + 126^2x^4 +...)^5 +...`
	PROG	`(PARI) {a(n)=polcoeff(sum(m=0, n, sum(k=0, m, binomial(m, k)^2x^k)^mx^m)+xO(x^n), n)}` `(PARI) {a(n)=polcoeff(sum(m=0, n, x^m(1-x+xO(x^n))^(2m^2+m)sum(k=0, n-m+1, binomial(m+k, k)^2x^k+x*O(x^n))^m), n)}`
	CROSSREFS	`Cf. A183165, A184355, A184356.` `Sequence in context: A161798 A134091 A219614 * A032331 A049371 A032079` `Adjacent sequences: A183163 A183164 A183165 * A183167 A183168 A183169`
	KEYWORD	`nonn`
	AUTHOR	`Paul D. Hanna, Dec 30 2010`
	STATUS	`approved`

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Last modified May 25 06:36 EDT 2013. Contains 225644 sequences.