A246840 -id:A246840 - cp's OEIS frontend

A181665 G.f. satisfies: A(x) = Sum_{n>=0} x^n[Sum_{k=0..n} C(n,k)^2 x^k* A(x)^k].

1, 1, 2, 6, 17, 51, 161, 519, 1707, 5711, 19358, 66342, 229505, 800333, 2810370, 9928806, 35266403, 125863071, 451119566, 1623142622, 5860507205, 21227095355, 77108788287, 280847802645, 1025416658863, 3752414144071, 13760368353098

Offset: 0

Paul D. Hanna, Jan 31 2011

Compare g.f. to the g.f. M(x) of Motzkin numbers:

M(x) = Sum_{n>=0} x^n*[Sum_{k=0..n} C(n,k) * x^k*M(x)^k].

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 17*x^4 + 51*x^5 + 161*x^6 + ...
where g.f. A(x) satisfies:
(1) A(x) = 1 + x*(1 + x*A(x)) + x^2*(1 + 4*x*A(x) + x^2*A(x)^2) + x^3*(1 + 9*x*A(x) + 9*x^2*A(x)^2 + x^3*A(x)^3) + x^4*(1 + 16*x*A(x) + 36*x^2*A(x)^2 + 16*x^3*A(x)^3 + x^4*A(x)^4) + ...
(2) A(x) = 1/(1-x) + x^2*A(x)*(1+x)/(1-x)^3 + x^4*A(x)^2*(1+4*x+x^2)/(1-x)^5  + x^6*A(x)^3*(1+9*x+9*x^2+x^3)/(1-x)^7 + ...
(3) A(x) = 1/(1-x-x^2*A(x)) + 2*x^3*A(x)/(1-x-x^2*A(x))^3 + 6*x^6*A(x)^2/(1-x-x^2*A(x))^5 + 20*x^9*A(x)^3/(1-x-x^2*A(x))^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A183876, A246840, A246861.

max = 27; se = 1/x*InverseSeries[ Series[ x/(x + x^2 + Sqrt[1 + 4*x^3]), {x, 0, max}], x]; CoefficientList[se, x] (* Jean-François Alcover, Mar 06 2013 *)

{a(n)=polcoeff((1/x)*serreverse(x/(x + x^2 + sqrt(1+4*x^3+O(x^(n+2))))),n)}

{a(n)=local(A=1+x);for(i=1,n,A=x*A+x^2*A^2+sqrt(1 + 4*x^3*A^3+x*O(x^n)));polcoeff(A,n)}

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,x^m*sum(k=0,m,binomial(m,k)^2*x^k*(A+x*O(x^n))^k)));polcoeff(A,n)}

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,x^m*(1-x*A)^(2*m+1)*sum(k=0,n,binomial(m+k,k)^2*x^k*(A+x^2*O(x^n))^k)));polcoeff(A,n)}

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n\2,x^(2*m)*(A+x*O(x^n))^m*sum(k=0,n,binomial(m+k,k)^2*x^k)));polcoeff(A,n)}

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n\2,x^(2*m)*A^m/(1-x+x*O(x^n))^(2*m+1)*sum(k=0,m,binomial(m,k)^2*x^k)));polcoeff(A,n)}

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\3,(2*m)!/m!^2*x^(3*m)*A^m/(1-x-x^2*A+x*O(x^n))^(2*m+1))); polcoeff(A, n)}

G.f. A(x) satisfies:
(1) A(x) = x*A(x) + x^2*A(x)^2 + sqrt(1 + 4*x^3*A(x)^3);
(2) A(x) = (1/x)*Series_Reversion[x/(x + x^2 + sqrt(1+4*x^3))];
(3) A(x) = -1 + x*A(x) + x^2*A(x)^2 + 2/C(-x^3*A(x)^3), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108);
(4) A(x) = Sum_{n>=0} x^n*(1 - x*A(x))^(2*n+1) * [Sum_{k>=0} C(n+k,k)^2 *x^k*A(x)^k];
(5) A(x) = Sum_{n>=0} x^(2n)*A(x)^n*[Sum_{k>=0} C(n+k,k)^2*x^k];
(6) A(x) = Sum_{n>=0} x^(2n)*A(x)^n*[Sum_{k=0..n} C(n,k)^2*x^k] /(1-x)^(2n+1);
(7) A(x) = Sum_{n>=0} (2n)!/n!^2 * x^(3n)*A(x)^n/(1-x-x^2*A(x))^(2n+1).
a(n) ~ sqrt(3*s^3/(-1 + 3*r + r^3 + 8*r^6*s^3 - 6*r^4*s*(1+2*s) + 3*r^2*(2*s-1))) / (sqrt(Pi)*n^(3/2)*r^(n-3/2)), where r = 0.25811980810324170407..., s = 2.3904081948888478693... are roots of the system of equations r + 2*r^2*s + (6*r^3*s^2)/sqrt(1 + 4*r^3*s^3) = 1, r*s + r^2*s^2 + sqrt(1 + 4*r^3*s^3) = s. - Vaclav Kotesovec, Mar 07 2014

A246883 Expansion of Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(3*k).

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 10, 17, 27, 46, 86, 165, 308, 558, 1006, 1841, 3421, 6383, 11863, 21966, 40697, 75662, 141099, 263429, 491778, 918104, 1715259, 3208078, 6005818, 11250198, 21082487, 39524241, 74135187, 139128897, 261228200, 490682127, 922015964, 1733127107, 3258939997, 6130162494, 11534742080

Offset: 0

Paul D. Hanna, Sep 06 2014

nonn

Limit a(n)/a(n+1) = t^2 = 0.524888598656404... (A072223) where t is the positive real root of 1 - x - x^4 = 0.

Diagonal of the rational function 1 / ((1-x)*(1-y) - (x*y)^4). - Seiichi Manyama, Apr 29 2025

			G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 5*x^5 + 10*x^6 + 17*x^7 +...
where, by definition,
A(x) = 1 + x*(1 + x^3) + x^2*(1 + 2^2*x^3 + x^6)
+ x^3*(1 + 3^2*x^3 + 3^2*x^6 + x^9)
+ x^4*(1 + 4^2*x^3 + 6^2*x^6 + 4^2*x^9 + x^12)
+ x^5*(1 + 5^2*x^3 + 10^2*x^6 + 10^2*x^9 + 5^2*x^12 + x^15) +...
which is also given by the series identity:
A(x) = 1/(1-x+x^4) + 2*x^4/(1-x+x^4)^3 + 6*x^8/(1-x+x^4)^5 + 20*x^12/(1-x+x^4)^7 + 70*x^16/(1-x+x^4)^9 + 252*x^20/(1-x+x^4)^11 + 924*x^24/(1-x+x^4)^13 +...
The logarithm of the g.f. begins:
log(A(x)) = x*(1 + x^3) + x^2*(1 + 6*x^3 + x^6)/2
+ x^3*(1 + 15*x^3 + 15*x^6 + x^9)/3
+ x^4*(1 + 28*x^3 + 70*x^6 + 28*x^9 + x^12)/4
+ x^5*(1 + 45*x^3 + 210*x^6 + 210*x^9 + 45*x^12 + x^15)/5 +...
more explicitly,
log(A(x)) = x + x^2/2 + x^3/3 + 5*x^4/4 + 16*x^5/5 + 31*x^6/6 + 50*x^7/7 + 77*x^8/8 + 145*x^9/9 + 306*x^10/10 + 628*x^11/11 + 1199*x^12/12 +...
where the logarithmic derivative equals
A'(x)/A(x) = (1-x+4*x^3+5*x^4-4*x^7)/((1-x+2*x^2+x^4)*(1-x-2*x^2+x^4)).

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.

Cf. A072223, A181665, A246840, A246884, A248193.

CoefficientList[Series[1/Sqrt[(1 - x + x^4)^2 - 4 x^4], {x, 0, 40}], x] (* Michael De Vlieger, Sep 10 2021 *)

/* By definition: */
{a(n)=local(A=1); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*x^(3*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From closed formula: */
{a(n)=local(A=1); A= 1/sqrt((1 - x + x^4)^2 - 4*x^4 +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From a series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n, (2*m)!/(m!)^2 * x^(4*m) / (1 - x + x^4 +x*O(x^n))^(2*m+1)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n, x^m*(1-x^3)^(2*m+1)*sum(k=0, n, binomial(m+k, k)^2*x^(3*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n\4, x^(4*m)*sum(k=0, n-4*m, binomial(m+k, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n\4, x^(4*m) * sum(k=0, m, binomial(m, k)^2*x^k) / (1-x +x*O(x^n))^(2*m+1) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From exponential formula: */
{a(n)=local(A=1); A=exp(sum(m=1, n, sum(k=0, m, binomial(2*m, 2*k)*x^(3*k)) * x^m/m) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From formula for a(n): */
{a(n)=sum(k=0, n\3, binomial(n-3*k, k)^2)}
for(n=0, 40, print1(a(n), ", "))

G.f.: Sum_{n>=0} (2*n)!/(n!)^2 * x^(4*n) / (1 - x + x^4)^(2*n+1). - Paul D. Hanna, Oct 15 2014
G.f.: Sum_{n>=0} x^n * [Sum_{k>=0} C(n+k,k)^2 * x^(3*k)] * (1-x^3)^(2*n+1).
G.f.: Sum_{n>=0} x^(4*n) * [Sum_{k>=0} C(n+k,k)^2 * x^k].
G.f.: Sum_{n>=0} x^(4*n) * [Sum_{k=0..n} C(n,k)^2 * x^k] /(1-x)^(2n+1).
G.f.: exp( Sum_{n>=1} (x^n/n) * Sum_{k=0..n} C(2*n,2*k) * x^(3*k) ).
G.f.: 1 / sqrt((1 - x + 2*x^2 + x^4)*(1 - x - 2*x^2 + x^4)).
G.f.: 1 / sqrt((1 - x + x^4)^2 - 4*x^4).
G.f.: 1 / sqrt((1 - x - x^4)^2 - 4*x^5).
a(n) = Sum_{k=0..[n/3]} C(n-3*k, k)^2.
n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + 2*(n-2)*a(n-4) + (2*n-5)*a(n-5) - (n-4)*a(n-8). - Seiichi Manyama, Aug 10 2024

A246884 Expansion of Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(4*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 5, 10, 17, 26, 38, 59, 101, 182, 326, 564, 945, 1566, 2622, 4476, 7750, 13455, 23231, 39837, 68101, 116611, 200526, 346137, 598438, 1034227, 1785400, 3080418, 5317009, 9187567, 15893830, 27515434, 47647774, 82513447, 142902640, 247553410, 429020710, 743846284

Offset: 0

Paul D. Hanna, Sep 06 2014

nonn

Limit a(n)/a(n+1) = t^2 = 0.569840290998053... where t = A075778 is the positive real root of 1 - x - x^5 = 0.

Diagonal of the rational function 1 / ((1-x)*(1-y) - (x*y)^5). - Seiichi Manyama, Apr 29 2025

			G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 5*x^6 + 10*x^7 + 17*x^8 +...
where, by definition,
A(x) = 1 + x*(1 + x^4) + x^2*(1 + 2^2*x^4 + x^8)
+ x^3*(1 + 3^2*x^4 + 3^2*x^8 + x^12)
+ x^4*(1 + 4^2*x^4 + 6^2*x^8 + 4^2*x^12 + x^16)
+ x^5*(1 + 5^2*x^4 + 10^2*x^8 + 10^2*x^12 + 5^2*x^16 + x^20) +...
which is also given by the series identity:
A(x) = 1/(1-x+x^5) + 2*x^5/(1-x+x^5)^3 + 6*x^10/(1-x+x^5)^5 + 20*x^15/(1-x+x^5)^7 + 70*x^20/(1-x+x^5)^9 + 252*x^25/(1-x+x^5)^11 + 924*x^30/(1-x+x^5)^13 +...
The logarithm of the g.f. begins:
log(A(x)) = x*(1 + x^4) + x^2*(1 + 6*x^4 + x^8)/2
+ x^3*(1 + 15*x^4 + 15*x^8 + x^12)/3
+ x^4*(1 + 28*x^4 + 70*x^8 + 28*x^12 + x^16)/4
+ x^5*(1 + 45*x^4 + 210*x^8 + 210*x^12 + 45*x^16 + x^20)/5 +...
more explicitly,
log(A(x)) = x + x^2/2 + x^3/3 + x^4/4 + 6*x^5/5 + 19*x^6/6 + 36*x^7/7 + 57*x^8/8 + 82*x^9/9 + 116*x^10/10 + 199*x^11/11 + 391*x^12/12 +...
where the logarithmic derivative equals
A'(x)/A(x) = (1-x+5*x^4+6*x^5-5*x^9)/((1+x+x^2)*(1-2*x+x^2-x^3)*(1-x+2*x^3-x^5)).

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.

Cf. A075778, A181665, A246840, A246883, A248193.

CoefficientList[Series[1/Sqrt[(1 - x + x^5)^2 - 4 x^5], {x, 0, 41}], x] (* Michael De Vlieger, Sep 10 2021 *)

/* By definition: */
{a(n)=local(A=1); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*x^(4*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From closed formula: */
{a(n)=local(A=1); A= 1/sqrt((1 - x - x^5)^2 - 4*x^6 +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From a series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n, (2*m)!/(m!)^2 * x^(5*m) / (1 - x + x^5 +x*O(x^n))^(2*m+1)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n, x^m*(1-x^4)^(2*m+1)*sum(k=0, n, binomial(m+k, k)^2*x^(4*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n\5, x^(5*m)*sum(k=0, n-4*m, binomial(m+k, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n\5, x^(5*m) * sum(k=0, m, binomial(m, k)^2*x^k) / (1-x +x*O(x^n))^(2*m+1) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From exponential formula: */
{a(n)=local(A=1); A=exp(sum(m=1, n, sum(k=0, m, binomial(2*m, 2*k)*x^(4*k)) * x^m/m) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From exponential formula: */
{a(n)=local(A=1); A=exp(sum(m=1, n, ((1+x^2)^(2*m) + (1-x^2)^(2*m))/2 * x^m/m) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From formula for a(n): */
{a(n)=sum(k=0, n\4, binomial(n-4*k, k)^2)}
for(n=0, 40, print1(a(n), ", "))

G.f.: Sum_{n>=0} (2*n)!/(n!)^2 * x^(5*n) / (1 - x + x^5)^(2*n+1). - Paul D. Hanna, Oct 15 2014
G.f.: Sum_{n>=0} x^n * [Sum_{k>=0} C(n+k,k)^2 * x^(4*k)] * (1-x^4)^(2*n+1).
G.f.: Sum_{n>=0} x^(5*n) * [Sum_{k>=0} C(n+k,k)^2 * x^k].
G.f.: Sum_{n>=0} x^(5*n) * [Sum_{k=0..n} C(n,k)^2 * x^k] /(1-x)^(2n+1).
G.f.: exp( Sum_{n>=1} (x^n/n) * Sum_{k=0..n} C(2*n,2*k) * x^(4*k) ).
G.f.: exp( Sum_{n>=1} (x^n/n) * ((1+x^2)^(2*n) + (1-x^2)^(2*n))/2 ).
G.f.: 1 / sqrt((1 - x - 2*x^3 - x^5)*(1 - x + 2*x^3 - x^5)).
G.f.: 1 / sqrt((1 - x - x^5)^2 - 4*x^6).
G.f.: 1 / sqrt((1 - x + x^5)^2 - 4*x^5).
a(n) = Sum_{k=0..[n/4]} C(n-4*k, k)^2.
n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + (2*n-5)*a(n-5) + 2*(n-3)*a(n-6) - (n-5)*a(n-10). - Seiichi Manyama, Aug 10 2024

A375278 Expansion of 1/((1 - x - x^3)^2 - 4*x^4).

Original entry on oeis.org

1, 2, 3, 6, 15, 34, 70, 146, 317, 690, 1480, 3162, 6788, 14608, 31395, 67392, 144701, 310854, 667793, 1434310, 3080542, 6616676, 14212315, 30526804, 65567936, 140832740, 302495240, 649730544, 1395554885, 2997508382, 6438345511, 13828920758, 29703127299

Offset: 0

Seiichi Manyama, Aug 09 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (2,-1,2,2,0,-1).

Cf. A182890, A375283, A375285.

Cf. A246840, A375279.

my(N=40, x='x+O('x^N)); Vec(1/((1-x-x^3)^2-4*x^4))

a(n) = sum(k=0, n\3, binomial(2*n-4*k+2, 2*k+1))/2;

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) + 2*a(n-4) - a(n-6).

a(n) = (1/2) * Sum_{k=0..floor(n/3)} binomial(2*n-4*k+2,2*k+1).

A248193 Expansion of Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(5*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 5, 10, 17, 26, 37, 51, 74, 118, 201, 347, 586, 955, 1509, 2351, 3682, 5871, 9545, 15700, 25851, 42292, 68606, 110635, 178190, 287852, 467313, 761957, 1245011, 2033856, 3317230, 5401332, 8787539, 14301168, 23301005, 38016585, 62090615, 101457357, 165778774

Offset: 0

Paul D. Hanna, Oct 03 2014

nonn

Limit a(n)/a(n+1) = t^2 = 0.6054234235718265... where t is the positive real root of 1 - x - x^6 = 0.

Diagonal of the rational function 1 / ((1-x)*(1-y) - (x*y)^6). - Seiichi Manyama, Apr 29 2025

			G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + 2*x^6 + 5*x^7 + 10*x^8 +...
where, by definition,
A(x) = 1 + x*(1 + x^5) + x^2*(1 + 2^2*x^5 + x^10)
+ x^3*(1 + 3^2*x^5 + 3^2*x^10 + x^15)
+ x^4*(1 + 4^2*x^5 + 6^2*x^10 + 4^2*x^15 + x^20)
+ x^5*(1 + 5^2*x^5 + 10^2*x^10 + 10^2*x^15 + 5^2*x^20 + x^25) +...
which is also given by the series identity:
A(x) = 1/(1-x+x^6) + 2*x^6/(1-x+x^6)^3 + 6*x^12/(1-x+x^6)^5 + 20*x^18/(1-x+x^6)^7 + 70*x^24/(1-x+x^6)^9 + 252*x^30/(1-x+x^6)^11 + 924*x^36/(1-x+x^6)^13 +...
The logarithm of the g.f. begins:
log(A(x)) = x*(1 + x^5) + x^2*(1 + 6*x^5 + x^10)/2
+ x^3*(1 + 15*x^5 + 15*x^10 + x^15)/3
+ x^4*(1 + 28*x^5 + 70*x^10 + 28*x^15 + x^20)/4
+ x^5*(1 + 45*x^5 + 210*x^10 + 210*x^15 + 45*x^20 + x^25)/5 +...
more explicitly,
log(A(x)) = x + x^2/2 + x^3/3 + x^4/4 + x^5/5 + 7*x^6/6 + 22*x^7/7 + 41*x^8/8 + 64*x^9/9 + 91*x^10/10 + 122*x^11/11 + 163*x^12/12 +...
where the logarithmic derivative equals
A'(x)/A(x) = (1 - x + 6*x^5 + 7*x^6 - 6*x^11) / ((1 - x + 2*x^3 + x^6)*(1 - x - 2*x^3 + x^6)).

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.

Cf. A181665, A246840, A246883, A246884.

CoefficientList[Series[1 / Sqrt[(1-x+x^6)^2 - 4*x^6], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 03 2014 *)

/* By definition: */
{a(n)=local(A=1); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*x^(5*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))

/* From closed formula: */
{a(n)=local(A=1); A= 1/sqrt((1 - x + x^6)^2 - 4*x^6 +x*O(x^n)); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))

/* From a series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n, (2*m)!/(m!)^2 * x^(6*m) / (1 - x + x^6 +x*O(x^n))^(2*m+1)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n, x^m*(1-x^5)^(2*m+1)*sum(k=0, n, binomial(m+k, k)^2*x^(5*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))

/* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n\6, x^(6*m)*sum(k=0, n-5*m, binomial(m+k, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))

/* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n\6, x^(6*m) * sum(k=0, m, binomial(m, k)^2*x^k) / (1-x +x*O(x^n))^(2*m+1) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From exponential formula: */
{a(n)=local(A=1); A=exp(sum(m=1, n, sum(k=0, m, binomial(2*m, 2*k)*x^(5*k)) * x^m/m) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))

/* From formula for a(n): */
{a(n)=sum(k=0, n\5, binomial(n-5*k, k)^2)}
for(n=0, 50, print1(a(n), ", "))

G.f.: Sum_{n>=0} (2*n)!/(n!)^2 * x^(6*n) / (1 - x + x^6)^(2*n+1). - Paul D. Hanna, Oct 15 2014
G.f.: Sum_{n>=0} x^n * [Sum_{k>=0} C(n+k,k)^2 * x^(5*k)] * (1-x^5)^(2*n+1).
G.f.: Sum_{n>=0} x^(6*n) * [Sum_{k>=0} C(n+k,k)^2 * x^k].
G.f.: Sum_{n>=0} x^(6*n) * [Sum_{k=0..n} C(n,k)^2 * x^k] /(1-x)^(2n+1).
G.f.: exp( Sum_{n>=1} (x^n/n) * Sum_{k=0..n} C(2*n,2*k) * x^(5*k) ).
G.f.: 1 / sqrt((1 - x + 2*x^3 + x^6)*(1 - x - 2*x^3 + x^6)).
G.f.: 1 / sqrt((1 - x + x^6)^2 - 4*x^6).
G.f.: 1 / sqrt((1 - x - x^6)^2 - 4*x^7).
a(n) = Sum_{k=0..[n/5]} C(n-5*k, k)^2.
n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + 2*(n-3)*a(n-6) + (2*n-7)*a(n-7) - (n-6)*a(n-12). - Seiichi Manyama, Aug 10 2024

A375292 Expansion of 1/sqrt((1 - x + x^3)^2 + 4*x^4).

Original entry on oeis.org

1, 1, 1, 0, -3, -8, -14, -15, 1, 51, 146, 261, 286, -24, -1029, -2975, -5375, -5930, 591, 22014, 63886, 115947, 128183, -14595, -486466, -1413161, -2569868, -2840890, 361667, 10972167, 31861581, 57980426, 64018181, -8985428, -250991300, -727998021, -1324662165

Offset: 0

Seiichi Manyama, Aug 10 2024

sign

Cf. A375021, A375293.

Cf. A246840.

my(N=40, x='x+O('x^N)); Vec(1/sqrt((1-x+x^3)^2+4*x^4))

a(n) = sum(k=0, n\3, (-1)^k*binomial(n-2*k, k)^2);

n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) - (2*n-3)*a(n-3) - 2*(n-2)*a(n-4) - (n-3)*a(n-6).

a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n-2*k,k)^2.

A387507 a(n) = Sum_{k=0..floor(n/3)} 2^k * binomial(n-2*k,k)^2.

Original entry on oeis.org

1, 1, 1, 3, 9, 19, 37, 87, 217, 507, 1157, 2727, 6553, 15627, 37077, 88519, 212569, 510715, 1226853, 2952615, 7120921, 17192427, 41538293, 100458759, 243211865, 589313755, 1428931333, 3467193191, 8418640793, 20453853003, 49722339861, 120936710471

Offset: 0

Seiichi Manyama, Aug 31 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A001850, A108488, A387508.

Cf. A246840, A375292.

[(&+[2^k * Binomial(n-2*k, k)^2: k in [0..Floor(n/3)]]): n in [0..40]]; // Vincenzo Librandi, Sep 02 2025

Table[Sum[2^k*Binomial[n-2*k, k]^2,{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 02 2025 *)

a(n) = sum(k=0, n\3, 2^k*binomial(n-2*k, k)^2);

G.f.: 1/sqrt((1-x-2*x^3)^2 - 8*x^4).

A246861 G.f. satisfies: A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * (xA(x))^(2k).

Original entry on oeis.org

1, 1, 1, 2, 7, 21, 54, 141, 407, 1231, 3691, 10990, 33144, 101674, 314679, 977289, 3047527, 9557503, 30133759, 95390622, 302960929, 965282651, 3085146472, 9888455045, 31774215928, 102334358736, 330298415136, 1068242904256, 3461372341327, 11235251353747, 36527859658661

Offset: 0

Paul D. Hanna, Sep 05 2014

nonn

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 21*x^5 + 54*x^6 + 141*x^7 + ...
where the g.f. A = A(x) equals the binomial series:
A(x) = 1 + x*(1 + x^2*A^2) + x^2*(1 + 2^2*x^2*A^2 + x^4*A^4)
+ x^3*(1 + 3^2*x^2*A^2 + 3^2*x^4*A^4 + x^6*A^6)
+ x^4*(1 + 4^2*x^2*A^2 + 6^2*x^4*A^4 + 4^2*x^6*A^6 + x^8*A^8)
+ x^5*(1 + 5^2*x^2*A^2 + 10^2*x^4*A^4 + 10^2*x^6*A^6 + 5^2*x^8*A^8 + x^10*A^10) + ...
Let A = g.f. A(x), then the g.f. satisfies:
log(A(x)) = x*(1 + x^2*A^2) + x^2*(1 + 6*x^2*A^2 + x^4*A^4)/2
+ x^3*(1 + 15*x^2*A^2 + 15*x^4*A^4 + x^6*A^6)/3
+ x^4*(1 + 28*x^2*A^2 + 70*x^4*A^4 + 28*x^6*A^6 + x^8*A^8)/4
+ x^5*(1 + 45*x^2*A^2 + 210*x^4*A^4 + 210*x^6*A^6 + 45*x^8*A^8 + x^10*A^10)/5 + ...
RELATED SERIES:
A(x)^2 = 1 + 2*x + 3*x^2 + 6*x^3 + 19*x^4 + 60*x^5 + 168*x^6 + ...
A(x)^3 = 1 + 3*x + 6*x^2 + 13*x^3 + 39*x^4 + 126*x^5 + 376*x^6 + ...
A(x)^4 = 1 + 4*x + 10*x^2 + 24*x^3 + 71*x^4 + 232*x^5 + 726*x^6 + ...
A(x)^6 = 1 + 6*x + 21*x^2 + 62*x^3 + 192*x^4 + 642*x^5 + 2145*x^6 + ...
where 1 = (1-x)^2*A(x)^2 - 2*x^3*(1+x)*A(x)^4 + x^6*A(x)^6.
Let G(x) = x/Series_Reversion(x*A(x)), then G(x*A(x)) = A(x), where
G(x) = x + x^3 + 1 + 2*x^4 - 2*x^8 + 4*x^12 - 10*x^16 + 28*x^20 - 84*x^24 + ...
G(x) = x + x^3 + sqrt(1 + 4*x^4).

Cf. A246840, A181665.

/* By definition: */
{a(n)=local(A=1); for(i=1,n, A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*(x*A)^(2*k)) +x*O(x^n))); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))

/* (1) From a binomial series identity: */
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m * sum(k=0, n, binomial(m+k, k)^2*(x*A +x*O(x^n))^(2*k)) * (1-x^2*A^2)^(2*m+1) )); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))

/* (2) From a binomial series identity: */
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\2, x^(3*m)*(A +x*O(x^n))^(2*m)*sum(k=0, n, binomial(m+k, k)^2*x^k))); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))

/* (3) From a binomial series identity: */
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\2, x^(3*m)*A^(2*m) * sum(k=0, m, binomial(m, k)^2*x^k) / (1-x +x*O(x^n))^(2*m+1) )); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))

/* (4) From exponential series formula: */
{a(n)=local(A=1); for(i=1,n, A=exp(sum(m=1, n, ((1+x*A)^(2*m) + (1-x*A)^(2*m))/2 * x^m/m) +x*O(x^n))); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))

/* (6) From functional equation: */
{a(n)=local(A=1); for(i=1,n, A =  1 / sqrt((1 - x*(1 - x*A)^2) * (1 - x*(1 + x*A)^2)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* (7) From functional equation: */
{a(n)=local(A=1); for(i=1,n, A = x*A + x^3*A^3 + sqrt(1 + 4*x^4*A^4 +x*O(x^n)) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* (8) From explicit formula: */
{a(n)=local(A=1); A= 1/x * serreverse( x / (x + x^3 + sqrt(1 + 4*x^4 +x*O(x^n) )) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n * [Sum_{k>=0} C(n+k,k)^2 *(x*A(x))^(2*k)] * (1 - x^2*A(x)^2)^(2*n+1).
(2) A(x) = Sum_{n>=0} x^(3*n) * A(x)^(2*n) * [Sum_{k>=0} C(n+k,k)^2 * x^k].
(3) A(x) = Sum_{n>=0} x^(3*n) * A(x)^(2*n) * [Sum_{k=0..n} C(n,k)^2 * x^k] /(1-x)^(2n+1).
(4) A(x) = exp( Sum_{n>=1} (x^n/n) * Sum_{k=0..n} C(2*n,2*k) * (x*A(x))^(2*k) ).
(5) A(x) = exp( Sum_{n>=1} (x^n/n) * ((1 + x*A(x))^(2*n) + (1 - x*A(x))^(2*n))/2 ).
(6) A(x) = 1 / sqrt((1 - x*(1 - x*A(x))^2) * (1 - x*(1 + x*A(x))^2)).
(7) A(x) = x*A(x) + x^3*A(x)^3 + sqrt(1 + 4*x^4*A(x)^4).
(8) A(x) = 1/x * Series_Reversion( x / (x + x^3 + sqrt(1 + 4*x^4)) ).

A375565 a(n) = Sum_{k=0..floor(n/3)} (n-2k+1) binomial(n-2*k,k)^2.

Original entry on oeis.org

1, 2, 3, 6, 17, 42, 90, 194, 441, 1006, 2242, 4950, 10974, 24376, 53961, 119048, 262337, 577782, 1271117, 2792718, 6129342, 13441616, 29454517, 64492800, 141108878, 308542280, 674238780, 1472532300, 3214268735, 7012637490, 15292425923, 33333338466, 72627184389

Offset: 0

Seiichi Manyama, Oct 17 2024

nonn

Cf. A182884, A376735.

Cf. A246840.

A375565 := proc(n)
    add((n-2*k+1)*binomial(n-2*k,k)^2,k=0..floor(n/3)) ;
end proc:
seq(A375565(n),n=0..80) ; # R. J. Mathar, Oct 17 2024

a(n) = sum(k=0, n\3, (n-2*k+1)*binomial(n-2*k, k)^2);

G.f.: (1-x-x^3)/((1-x-x^3)^2 - 4*x^4)^(3/2).

D-finite with recurrence 3*n*(n-1)*a(n) -(8*n-3)*(n-1)*a(n-1) +(7*n^2-14*n+8)*a(n-2) +(-8*n^2+3*n+23)*a(n-3) -2*n*(n+8)*a(n-4) +4*((n-1)^2)*a(n-5) +3*n*(n+2)*a(n-6) -2*n*(n-1)*a(n-7)=0. - R. J. Mathar, Oct 17 2024

A387510 a(n) = Sum_{k=0..floor(n/3)} 2^(n-2k) binomial(n-2*k,k)^2.

Original entry on oeis.org

1, 2, 4, 10, 32, 104, 324, 1000, 3136, 9992, 32064, 103168, 332816, 1077152, 3497024, 11381920, 37121280, 121285760, 396922944, 1300906112, 4269367296, 14028169344, 46143475712, 151932559360, 500710965504, 1651533562368, 5451595506688, 18008220715520

Offset: 0

Seiichi Manyama, Aug 31 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A387507, A387509.

Cf. A246840, A387513.

[(&+[2^(n-2*k) * Binomial(n-2*k, k)^2: k in [0..Floor(n/3)]]): n in [0..40]]; // Vincenzo Librandi, Sep 01 2025

Table[Sum[2^(n-2*k)*Binomial[n-2*k,k]^2,{k,0,Floor[n/3]}],{n,0,30}] (* Vincenzo Librandi, Sep 01 2025 *)

a(n) = sum(k=0, n\3, 2^(n-2*k)*binomial(n-2*k, k)^2);

G.f.: 1/sqrt((1-2*x-2*x^3)^2 - 16*x^4).

cp's OEIS Frontend

A181665 G.f. satisfies: A(x) = Sum_{n>=0} x^n*[Sum_{k=0..n} C(n,k)^2 *x^k* A(x)^k].

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A246883 Expansion of Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(3*k).

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A246884 Expansion of Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(4*k).

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A375278 Expansion of 1/((1 - x - x^3)^2 - 4*x^4).

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A248193 Expansion of Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(5*k).

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A181665 G.f. satisfies: A(x) = Sum_{n>=0} x^n[Sum_{k=0..n} C(n,k)^2 x^k* A(x)^k].

A246861 G.f. satisfies: A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * (xA(x))^(2k).

A375565 a(n) = Sum_{k=0..floor(n/3)} (n-2k+1) binomial(n-2*k,k)^2.

A387510 a(n) = Sum_{k=0..floor(n/3)} 2^(n-2k) binomial(n-2*k,k)^2.