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A132455 a(n) = sum of the squares of the coefficients of x^n in x^(n-2*k)/A(x^2)^(n-2*k+1), as k varies from 0 to floor(n/2), with a(0)=1, where A(x) is the g.f. of this sequence. 1
1, 1, 2, 5, 11, 18, 30, 45, 80, 147, 330, 882, 1935, 3298, 4676, 7613, 18409, 53724, 141094, 293407, 473604, 639418, 1138534, 3086050, 9159520, 23937239, 54502764, 106105954, 167076383, 237933308, 491988696, 1749801789, 6074959380 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The definition is a variation of the following property of the Catalan numbers: A000108(n) = Sum_{k=0..[n/2]} ( [x^n] (x*C(x^2))^(n-2*k+1) )^2, where C(x) is the g.f. of the Catalan numbers.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..100

FORMULA

a(n) = Sum_{k=0..[n/2]} ( [x^n] (x/A(x^2))^(n-2*k+1) )^2 for n>0 with a(0)=1.

EXAMPLE

a(n) is the sum of squares of coefficients in x^(n-2*k)/A(x^2)^(n-2*k+1):

a(2) =  2 = 1^2 + (-1)^2;

a(3) =  5 = 1^2 + (-2)^2;

a(4) = 11 = 1^2 + (-3)^2 + (-1)^2;

a(5) = 18 = 1^2 + (-4)^2 + (-1)^2;

a(6) = 30 = 1^2 + (-5)^2 + 0^2 + (-2)^2;

a(7) = 45 = 1^2 + (-6)^2 + 2^2 + (-2)^2;

a(8) = 80 = 1^2 + (-7)^2 + 5^2 + (-1)^2 + (-2)^2;

a(9) =147 = 1^2 + (-8)^2 + 9^2 + 0^2 + 1^2; ...

as can be seen from the initial coefficients in x^n/A(x^2)^n:

A(x)........: 1,1,2,5,11,18,30,45,80,147,...

x^1/A(x^2)^1: 1,_-1,__-1,___-2,___-2, ......

x^2/A(x^2)^2: __1,_-2,___-1,___-2,_____1,...

x^3/A(x^2)^3: ____1,__-3,____0,___-1, ......

x^4/A(x^2)^4: ______1,___-4,____2,_____0,...

x^5/A(x^2)^5: _________1,___-5,____5, ......

x^6/A(x^2)^6: ____________1,___-6,_____9,...

x^7/A(x^2)^7: _______________1,___-7, ......

x^8/A(x^2)^8: __________________1,____-8,...

x^9/A(x^2)^9: _____________________1, ......

x^10/A(x^2)^10: _______________________1,...

PROG

(PARI) {a(n)=if(n==0, 1, sum(k=0, n\2, polcoeff(x^(n-2*k)*(sum(j=0, k, a(j)*x^(2*j)) +x*O(x^n))^(-n+2*k-1), n)^2))}

CROSSREFS

Cf. A095892 (variant); A000108.

Sequence in context: A264724 A298375 A260037 * A132459 A308517 A101057

Adjacent sequences:  A132452 A132453 A132454 * A132456 A132457 A132458

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 21 2007

STATUS

approved

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Last modified October 19 15:50 EDT 2019. Contains 328223 sequences. (Running on oeis4.)