A162249 a(n) = sum of the squares of the coefficients of x^(2k) in A(x^2)^{2*(n-2k)+1}, as k varies from 0 to floor(n/2), with a(0)=1.

1, 1, 2, 10, 30, 131, 582, 3196, 13986, 70100, 336416, 1723518, 8487202, 44468780, 228236112, 1241788448, 6421700878, 34682391148, 182473774272, 993091141104, 5264377375260, 28721435063423, 153844326005054, 843854383167940

Offset

0,3

Comments

This is a variant of the following property of the Catalan sequence:

A000108(n) = sum of the squares of the coefficients of x^(2k) in Catalan(x^2)^{n-2k+1}, as k varies from 0 to floor(n/2) where Catalan(x) = g.f. of A000108.

Links

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

Formula

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

Example

To illustrate the recurrence, list coefficients of A(x^2)^(2n+1):
A^1: . 1,... 1,... 2,... 10,... 30,... 131,.......;
A^3: .... 1,... 3,... 9,... 43,... 168,... 735, ...;
A^5: ....... 1,... 5,... 20,... 100,... 455,.......;
A^7: .......... 1,... 7,... 35,... 189,... 959, ...;
A^9: ............. 1,... 9,... 54,... 318,.......;
A^11: ............... 1,... 11,... 77,... 495, ...;
A^13: .................. 1,... 13,... 104,.......;
A^15: ..................... 1,... 15,... 135, ...;
A^17: ........................ 1,... 17,.......;
A^19: ........................... 1,... 19, ...;
A^21: .............................. 1,.......;
A^23: ................................. 1, ...;...
then sum the squares of the coefficients in each column:
a(0) = 1^2 = 1;
a(1) = 1^2 = 1;
a(2) = 1^2 + 1^2 = 2;
a(3) = 3^2 + 1^2 = 10;
a(4) = 2^2 + 5^2 + 1^2 = 30;
a(5) = 9^2 + 7^2 + 1^2 = 131;
a(6) = 10^2 + 20^2 + 8^2 + 1^2 = 582;
a(7) = 43^2 + 35^2 + 11^2 + 1^2 = 3196;
a(8) = 30^2 + 100^2 + 54^2 + 13^2 + 1^2 = 13986;
a(9) = 168^2 + 189^2 + 77^2 + 15^2 + 1^2 = 70100.

Prog

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

Crossrefs

Cf. A095892.

Sequence in context: A285956 A219667 A192381 * A371484 A268688 A281069

Adjacent sequences: A162246 A162247 A162248 * A162250 A162251 A162252

Keyword

nonn

Author

Paul D. Hanna, Jun 28 2009

Extensions

Comment corrected by Paul D. Hanna, Jul 05 2009

Status

approved