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Search: a216490 -id:a216490
Displaying 1-4 of 4 results found.
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A112806
Expansion of solution of functional equation.
+10
4
1, 1, 2, 6, 21, 79, 312, 1277, 5369, 23049, 100612, 445214, 1992606, 9004260, 41025315, 188259072, 869305315, 4036286518, 18832973733, 88259024068, 415252542641, 1960718710035, 9288106921038, 44129146527731
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
Gi-Sang Cheon, S.-T. Jin, L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Available online 30 March 2015.
FORMULA
Given g.f. A(x), then series reversion of B(x)=x*A(x^3) is -B(-x).
Given g.f. A(x), then y=x*A(x^3) satisfies y=x+(xy)^2/(1-(xy)^3).
G.f. satisfies: A(x) = 1 + x*A(x)^2/(1 - x^2*A(x)^3). - Paul D. Hanna, Jun 06 2012
G.f. satisfies: A(x) = 1/A(-x*A(x)^3); note that the Catalan function C(x) = 1 + x*C(x)^2 (A000108) also satisfies this condition. - Paul D. Hanna, Jun 06 2012
a(n) = Sum_{i=0..n/2}((binomial(n+2*i+1,i)*Sum_{k=0..n-2*i}(binomial(k,n-k-2*i)*(-1)^(n-k)*binomial(n+k+2*i,k)))/(n+2*i+1)). - Vladimir Kruchinin, Mar 07 2016
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1,k) * binomial(2*n-k+1,n-2*k)/(2*n-k+1). - Seiichi Manyama, Aug 28 2023
PROG
(PARI) {a(n)=local(A); if(n<0, 0, A=x+O(x^4); for(k=1, n, A=x+subst(x^2/(1-x^3), x, x*A)); polcoeff(A, 3*n+1))}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+x*A^2/(1-x^2*A^3)); polcoeff(A, n)} \\ Paul D. Hanna, Jun 06 2012
(Maxima)
a(n):=sum((binomial(n+2*i+1, i)*sum(binomial(k, n-k-2*i)*(-1)^(n-k)*binomial(n+k+2*i, k), k, 0, n-2*i))/(n+2*i+1), i, 0, n/2); /* Vladimir Kruchinin, Mar 07 2016 */
CROSSREFS
Cf. A004148, A216490.
KEYWORD
nonn
AUTHOR
Michael Somos, Sep 20 2005
STATUS
approved
A216493
G.f. satisfies A(x) = 1 + x*A(x)^3 + x^5*A(x)^13.
+10
1
1, 1, 3, 12, 55, 274, 1444, 7923, 44803, 259325, 1529008, 9151327, 55454164, 339543312, 2097460255, 13055579858, 81803671623, 515552408141, 3265924761595, 20784056808550, 132812937949820, 851847261569025, 5482066256568375, 35388168141000935, 229081418808206500, 1486757986305948780, 9672120691595571320
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..26.
FORMULA
G.f. satisfies: A(x) = 1/A(-x*A(x)^5); note that the g.f. of A001764, G(x) = 1 + x*G(x)^3, also satisfies this condition.
a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k,k) * binomial(3*n-2*k+1,n-4*k)/(3*n-2*k+1). - Seiichi Manyama, Aug 28 2023
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 274*x^5 + 1444*x^6 + 7923*x^7 +...
Related expansions:
A(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1431*x^5 + 7806*x^6 + 43893*x^7 +...
A(x)^13 = 1 + 13*x + 117*x^2 + 910*x^3 + 6578*x^4 + 45643*x^5 + 309127*x^6 +...
Given (1) A(x) = 1 + x*A(x)^3 + x^5*A(x)^13,
suppose (2) A(x) = 1/A(-x*A(x)^5),
then substituting x in (1) with -x*A(x)^5 yields:
1/A(x) = 1 - x*A(x)^5/A(x)^3 - x^5*A(x)^25/A(x)^13,
which illustrates that (2) is consistent with (1).
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A^3+x^5*A^13 +x*O(x^n)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A216494, A216490, A001764.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 07 2012
STATUS
approved
A228987
G.f. satisfies A(x) = 1 + x*A(x)^3 + x^3*A(x)^8.
+10
1
1, 1, 3, 13, 66, 364, 2116, 12768, 79222, 502297, 3240120, 21196593, 140295584, 937787728, 6321624862, 42926227470, 293350136170, 2015999854478, 13923926272607, 96598395025615, 672852440805930, 4703751150849738, 32991286060134402, 232091541493091566
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..500
FORMULA
G.f. satisfies: A(x) = 1/A(-x*A(x)^5); note that the function G(x) = 1 + x*G(x)^3 (g.f. of A001764) also satisfies this condition: G(x) = 1/G(-x*G(x)^5).
a(n) ~ sqrt((3 - r*s^2)/(2*Pi*(3 + 28*r^2*s^5))) / (4*n^(3/2)*r^(n + 1/2)), where r = 0.1331154541373089587498695338172936885734070972340... and s = 1.408602671059676188189711196409966797670750551605... are real roots of the system of equations 1 + r*s^3 + r^3*s^8 = s, 3*r*s^2 + 8*r^3*s^7 = 1. - Vaclav Kotesovec, Nov 22 2017
a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k,k) * binomial(3*n-k+1,n-2*k)/(3*n-k+1). - Seiichi Manyama, Aug 28 2023
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 66*x^4 + 364*x^5 + 2116*x^6 +...
Related expansions:
A(x)^3 = 1 + 3*x + 12*x^2 + 58*x^3 + 312*x^4 + 1788*x^5 + 10686*x^6 +...
A(x)^8 = 1 + 8*x + 52*x^2 + 328*x^3 + 2082*x^4 + 13384*x^5 + 87124*x^6 +...
Given (1) A(x) = 1 + x*A(x)^3 + x^3*A(x)^8,
suppose (2) A(x) = 1/A(-x*A(x)^5),
then substituting x in (1) with -x*A(x)^5 yields:
1/A(x) = 1 - x*A(x)^5/A(x)^3 - x^3*A(x)^15/A(x)^8,
which illustrates that (2) is consistent with (1).
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A^3+x^3*A^8 +x*O(x^n)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A216490, A112806.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 10 2013
STATUS
approved
A367056
G.f. satisfies A(x) = 1 + x*A(x)^2 + x^3*A(x).
+10
1
1, 1, 2, 6, 17, 52, 168, 561, 1922, 6719, 23871, 85938, 312823, 1149421, 4257460, 15880036, 59594517, 224856450, 852491806, 3245959002, 12407332166, 47592364107, 183139542306, 706794663136, 2735053815771, 10609811267757, 41251228784198
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..26.
FORMULA
G.f.: A(x) = 2 / (1-x^3+sqrt((1-x^3)^2-4*x)).
a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k+1,k) * binomial(2*n-5*k,n-3*k)/(n-2*k+1).
D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(-2*n+7)*a(n-3) +(n-8)*a(n-6)=0. - R. J. Mathar, Dec 04 2023
MAPLE
A367056 := proc(n)
add(binomial(n-2*k+1, k) * binomial(2*n-5*k, n-3*k)/(n-2*k+1), k=0..floor(n/3)) ;
end proc:
seq(A367056(n), n=0..70) ; # R. J. Mathar, Dec 04 2023
PROG
(PARI) a(n) = sum(k=0, n\3, binomial(n-2*k+1, k)*binomial(2*n-5*k, n-3*k)/(n-2*k+1));
CROSSREFS
Cf. A049140, A071969, A125305, A216490, A367042.
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Nov 04 2023
STATUS
approved
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