cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A366233 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(3*x*A(x)).

Original entry on oeis.org

1, 1, 8, 87, 1428, 31125, 847818, 27785205, 1065267864, 46802921769, 2319200977230, 127985909702409, 7785440359004916, 517616528753919933, 37344834374921549154, 2906043724955696034285, 242627026212699695954352, 21634774261037677172406609, 2052077586846349144929739542
Offset: 0

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Author

Paul D. Hanna, Oct 05 2023

Keywords

Comments

Related identities which hold formally for all Maclaurin series F(x):
(1) F(x) = (1/x) * Sum_{n>=1} n^(n-1) * x^n/n! * F(x)^n * exp(-n*x*F(x)),
(2) F(x) = (2/x) * Sum_{n>=1} n*(n+1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+1)*x*F(x)),
(3) F(x) = (3/x) * Sum_{n>=1} n*(n+2)^(n-2) * x^n/n! * F(x)^n * exp(-(n+2)*x*F(x)),
(4) F(x) = (4/x) * Sum_{n>=1} n*(n+3)^(n-2) * x^n/n! * F(x)^n * exp(-(n+3)*x*F(x)),
(5) F(x) = (k/x) * Sum_{n>=1} n*(n+k-1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+k-1)*x*F(x)) for all fixed nonzero k.

Examples

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 87*x^3/3! + 1428*x^4/4! + 31125*x^5/5! + 847818*x^6/6! + 27785205*x^7/7! + 1065267864*x^8/8! + ...
where A(x) satisfies A(x) = 1 + x*A(x) * exp(3*x*A(x))
also
A(x) = 1 + 1^0*x*A(x)*exp(+2*x*A(x))/1! + 2^1*x^2*A(x)^2*exp(+1*x*A(x))/2! + 3^2*x^3*A(x)^3*exp(-0*x*A(x))/3! + 4^3*x^4*A(x)^4*exp(-1*x*A(x))/4! + 5^4*x^5*A(x)^5*exp(-2*x*A(x))/5! + 6^5*x^6*A(x)^6*exp(-3*x*A(x))/6! + ...
and
A(x) = 1 + 4*1*4^(-1)*x*A(x)*exp(-1*x*A(x))/1! + 4*2*5^0*x^2*A(x)^2*exp(-2*x*A(x))/2! + 4*3*6^1*x^3*A(x)^3*exp(-3*x*A(x))/3! + 4*4*7^2*x^4*A(x)^4*exp(-4*x*A(x))/4! + 4*5*8^3*x^5*A(x)^5*exp(-5*x*A(x))/5! + ...
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 31*x^3/3! + 469*x^4/4! + 9681*x^5/5! + 254701*x^6/6! + ... + A212917(n)*x^n/n! + ...

Crossrefs

Cf. A161633, A366232, A366234, A366235.
Cf. A365773 (dual), A212917 (exp(x*A(x))).

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1 + x*A[x] * E^(3*x*A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0,nmax]! (* Vaclav Kotesovec, Oct 06 2023 *)
  • PARI
    /* a(n,m) = coefficient of x^n/n! in A(x)^m, here at m = 1 */
{a(n, m=1) = n!*sum(k=0, n, binomial(n+m, n-k)*m/(n+m) * 3^k * (n-k)^k/k!)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n) = my(A = (1/x) * serreverse( x/(1 + x*exp(3*x +O(x^(n+2)))) )); n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = n! * Sum_{k=0..n} binomial(n+1, n-k)/(n+1) * 3^k * (n-k)^k / k!.
Let A(x)^m = Sum_{n>=0} a(n,m) * x^n/n! then a(n,m) = n!*Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * 3^k * (n-k)^k/k!.
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + x*A(x) * exp(3*x*A(x)).
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(3*x)) ).
(3) A( x/(1 + x*exp(3*x)) ) = 1 + x*exp(3*x).
(4) A(x) = 1 + (m+1) * Sum_{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-3)*x*A(x)) for all fixed nonnegative m.
(4.a) A(x) = 1 + Sum_{n>=1} n^(n-1) * x^n/n! * A(x)^n * exp(-(n-3)*x*A(x)).
(4.b) A(x) = 1 + 2 * Sum_{n>=1} n*(n+1)^(n-2) * x^n/n! * A(x)^n * exp(-(n-2)*x*A(x)).
(4.c) A(x) = 1 + 3 * Sum_{n>=1} n*(n+2)^(n-2) * x^n/n! * A(x)^n * exp(-(n-1)*x*A(x)).
(4.d) A(x) = 1 + 4 * Sum_{n>=1} n*(n+3)^(n-2) * x^n/n! * A(x)^n * exp(-n*x*A(x)).
(4.e) A(x) = 1 + 5 * Sum_{n>=1} n*(n+4)^(n-2) * x^n/n! * A(x)^n * exp(-(n+1)*x*A(x)).
a(n) ~ 3^n * (1 + 2*LambertW(sqrt(3)/2))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(sqrt(3)/2)) * 2^(2*n + 2) * exp(n) * LambertW(sqrt(3)/2)^(2*n + 3/2)). - Vaclav Kotesovec, Oct 06 2023

A365770 Expansion of g.f. A(x,y) satisfying A(x,y) = 1 + x*A(x,y)/(1 - x*y * A(x,y))^2, as a triangle of coefficients T(n,k) of x^n*y^k in A(x,y), read by rows n >= 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 12, 20, 4, 0, 1, 20, 70, 50, 5, 0, 1, 30, 180, 280, 105, 6, 0, 1, 42, 385, 1050, 882, 196, 7, 0, 1, 56, 728, 3080, 4620, 2352, 336, 8, 0, 1, 72, 1260, 7644, 18018, 16632, 5544, 540, 9, 0, 1, 90, 2040, 16800, 57330, 84084, 51480, 11880, 825, 10, 0, 1, 110, 3135, 33660, 157080, 336336, 330330, 141570, 23595, 1210, 11, 0
Offset: 0

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Author

Paul D. Hanna, Oct 10 2023

Keywords

Comments

A365771(n) = T(2*n,n), the central terms.
A109081(n) = Sum_{k=0..n} T(n,k), the row sums.
A365772(n) = Sum_{k=0..n} T(n,k) * 2^k.
A365773(n) = Sum_{k=0..n} T(n,k) * 3^k.
A365774(n) = Sum_{k=0..n} T(n,k) * 4^k.
A365775(n) = Sum_{k=0..n} T(n,k) * 5^k.
Related identities which hold formally for all Maclaurin series F(x):
(1) F(x) = (1/x) * Sum{n>=1} n^(n-1) * x^n * F(x)^n / (1 + n*x*F(x))^(n+1),
(2) F(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n * F(x)^n / (1 + (n+1)*x*F(x))^(n+1),
(3) F(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n * F(x)^n / (1 + (n+2)*x*F(x))^(n+1),
(4) F(x) = (4/x) * Sum{n>=1} n*(n+3)^(n-2) * x^n * F(x)^n / (1 + (n+3)*x*F(x))^(n+1),
(5) F(x) = (k/x) * Sum{n>=1} n*(n+k-1)^(n-2) * x^n * F(x)^n / (1 + (n+k-1)*x*F(x))^(n+1) for all fixed nonzero k.

Examples

			G.f.: A(x,y) = 1 + x + (1 + 2*y)*x^2 + (1 + 6*y + 3*y^2)*x^3 + (1 + 12*y + 20*y^2 + 4*y^3)*x^4 + (1 + 20*y + 70*y^2 + 50*y^3 + 5*y^4)*x^5 + (1 + 30*y + 180*y^2 + 280*y^3 + 105*y^4 + 6*y^5)*x^6 + (1 + 42*y + 385*y^2 + 1050*y^3 + 882*y^4 + 196*y^5 + 7*y^6)*x^7 + (1 + 56*y + 728*y^2 + 3080*y^3 + 4620*y^4 + 2352*y^5 + 336*y^6 + 8*y^7)*x^8 + ...
where
A(x,y) = 1 + x*A(x,y)/(1 - x*y*A(x,y))^2.
Also,
A(x,y) = 1 + 1^0*x*A(x,y)/(1 + (1-y)*x*A(x,y))^2 + 2^1*x^2*A(x,y)^2/(1 + (2-y)*x*A(x,y))^3 + 3^2*x^3*A(x,y)^3/(1 + (3-y)*x*A(x,y))^4 + 4^3*x^4*A(x,y)^4/(1 + (4-y)*x*A(x,y))^5 + 5^4*x^5*A(x,y)^5/(1 + (5-y)*x*A(x,y))^6 + ...
and
A(x,y) = 1 + (1+y)*1*(1+y)^(-1)*x*A(x,y)/(1 + 1*x*A(x,y))^2 + (1+y)*2*(2+y)^0*x^2*A(x,y)^2/(1 + 2*x*A(x,y))^3 + (1+y)*3*(3+y)^1*x^3*A(x,y)^3/(1 + 3*x*A(x,y))^4 + (1+y)*4*(4+y)^2*x^4*A(x,y)^4/(1 + 4*x*A(x,y))^5 + ...
This triangle of coefficients of x^n*y^k in A(x,y) begins:
1;
1, 0;
1, 2, 0;
1, 6, 3, 0;
1, 12, 20, 4, 0;
1, 20, 70, 50, 5, 0;
1, 30, 180, 280, 105, 6, 0;
1, 42, 385, 1050, 882, 196, 7, 0;
1, 56, 728, 3080, 4620, 2352, 336, 8, 0;
1, 72, 1260, 7644, 18018, 16632, 5544, 540, 9, 0;
1, 90, 2040, 16800, 57330, 84084, 51480, 11880, 825, 10, 0; ...

Links

Crossrefs

Cf. A109081 (y=1), A365772 (y=2), A365773 (y=3), A365774 (y=4), A365775 (y=5).
Cf. A365771 (central terms).

Programs

  • PARI
    {T(n,k) = binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k)}
for(n=0,10, for(k=0,n, print1(T(n,k),", "));print(""))

Formula

T(n,k) = binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k).
G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k satisfies the following formulas.
(1) A(x,y) = 1 + x*A(x,y)/(1 - x*y*A(x,y))^2.
(2) A(x,y) = (1/x) * Series_Reversion( x/(1 + x/(1 - x*y)^2) ), where reversion is taken wrt x.
(3) A( x/(1 + x/(1 - x*y)^2), y) = 1 + x/(1 - x*y)^2.
(4) A(x,y) = 1 + (1+y) * Sum{n>=1} n*(n+y)^(n-2) * x^n * A(x,y)^n / (1 + n*x*A(x,y))^(n+1).
(5) A(x,y) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n * A(x,y)^n / (1 + (n+m-y)*x*A(x,y))^(n+1) for all fixed nonnegative m.
(5.a) A(x,y) = 1 + Sum{n>=1} n^(n-1) * x^n * A(x,y)^n / (1 + (n-y)*x*A(x,y))^(n+1).
(5.b) A(x,y) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n * A(x,y)^n / (1 + (n+1-y)*x*A(x,y))^(n+1).
(5.c) A(x,y) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n * A(x,y)^n / (1 + (n+2-y)*x*A(x,y))^(n+1).
(5.d) A(x,y) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n * A(x,y)^n / (1 + (n+3-y)*x*A(x,y))^(n+1).

A365772 Expansion of g.f. A(x) satisfying A(x) = 1 + x*A(x)/(1 - 2*x*A(x))^2.

Original entry on oeis.org

1, 1, 5, 25, 137, 801, 4893, 30857, 199377, 1313089, 8782389, 59491257, 407308377, 2814044897, 19594237133, 137364464681, 968743846561, 6868059398273, 48921561805413, 349942779608153, 2512722402972457, 18104571857859233, 130856263145140861, 948520413875412681
Offset: 0

Views

Author

Paul D. Hanna, Oct 04 2023

Keywords

Comments

Related identities which hold formally for all Maclaurin series F(x):
(1) F(x) = (1/x) * Sum{n>=1} n^(n-1) * x^n * F(x)^n / (1 + n*x*F(x))^(n+1),
(2) F(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n * F(x)^n / (1 + (n+1)*x*F(x))^(n+1),
(3) F(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n * F(x)^n / (1 + (n+2)*x*F(x))^(n+1),
(4) F(x) = (4/x) * Sum{n>=1} n*(n+3)^(n-2) * x^n * F(x)^n / (1 + (n+3)*x*F(x))^(n+1),
(5) F(x) = (k/x) * Sum{n>=1} n*(n+k-1)^(n-2) * x^n * F(x)^n / (1 + (n+k-1)*x*F(x))^(n+1) for all fixed nonzero k.

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 25*x^3 + 137*x^4 + 801*x^5 + 4893*x^6 + 30857*x^7 + 199377*x^8 + 1313089*x^9 + 8782389*x^10 + ...
where A(x) satisfies A(x) = 1 + x*A(x)/(1 - 2*x*A(x))^2
also
A(x) = 1 + x*A(x)/(1 + (-1)*x*A(x))^2 + 2*x^2*A(x)^2/(1 + 0*x*A(x))^3 + 3^2*x^3*A(x)^3/(1 + 1*x*A(x))^4 + 4^3*x^4*A(x)^4/(1 + 2*x*A(x))^5 + 5^4*x^5*A(x)^5/(1 + 3*x*A(x))^6 + 6^5*x^6*A(x)^6/(1 + 4*x*A(x))^7 + ...
and
A(x) = 1 + 3*1*3^(-1)*x*A(x)/(1 + 1*x*A(x))^2 + 3*2*4^0*x^2*A(x)^2/(1 + 2*x*A(x))^3 + 3*3*5^1*x^3*A(x)^3/(1 + 3*x*A(x))^4 + 3*4*6^2*x^4*A(x)^4/(1 + 4*x*A(x))^5 + 3*5*7^3*x^5*A(x)^5/(1 + 5*x*A(x))^6 + ...

Links

Crossrefs

Cf. A365770, A109081, A365773, A365774, A365775.
Cf. A366232 (dual).

Programs

  • Mathematica
    nmax = 30; A[] = 0; Do[A[x] = 1 + x*A[x]/(1 - 2*x*A[x])^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Oct 05 2023 *)
  • PARI
    {a(n) = sum(k=0, n, binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 2^k)}
for(n=0,30, print1(a(n),", "))
  • PARI
    {a(n) = my(A = (1/x) * serreverse( x/(1 + x/(1 - 2*x +O(x^(n+2)) )^2) ) ); polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))

Formula

a(n) = Sum{k=0..n} binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 2^k.
Let A(x)^m = Sum_{n>=0} a(n,m) * x^n then a(n,m) = Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * binomial(2*n-k-1, k) * 2^k.
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*A(x)/(1 - 2*x*A(x))^2.
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x/(1 - 2*x)^2) ).
(3) A( x/(1 + x/(1 - 2*x)^2) ) = 1 + x/(1 - 2*x)^2.
(4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n * A(x)^n / (1 + (n+m-2)*x*A(x))^(n+1) for all fixed nonnegative m.
(4.a) A(x) = 1 + Sum{n>=1} n^(n-1) * x^n * A(x)^n / (1 + (n-2)*x*A(x))^(n+1).
(4.b) A(x) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n * A(x)^n / (1 + (n-1)*x*A(x))^(n+1).
(4.c) A(x) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n * A(x)^n / (1 + n*x*A(x))^(n+1).
(4.d) A(x) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n * A(x)^n / (1 + (n+1)*x*A(x))^(n+1).
a(n) ~ 7^(n + 3/2) * sqrt(3/((-1916 + (1833997600 - 95194848*sqrt(69))^(1/3) + 2^(5/3)*(57312425 + 2974839*sqrt(69))^(1/3))*Pi)) / (2 * n^(3/2) * (1 - 53*(2/(3*(-45 + 161*sqrt(69))))^(1/3) + ((-45 + 161*sqrt(69))/2)^(1/3)/3^(2/3))^n). - Vaclav Kotesovec, Oct 05 2023

A365774 Expansion of g.f. A(x) satisfying A(x) = 1 + x*A(x)/(1 - 4*x*A(x))^2.

Original entry on oeis.org

1, 1, 9, 73, 625, 5681, 53945, 528697, 5307489, 54298849, 564079337, 5934390441, 63098046929, 676976915473, 7319925023897, 79684985945753, 872620958369473, 9606337027601345, 106249046704511945, 1180096759408431881, 13156993620315230001, 147193406523115480049
Offset: 0

Views

Author

Paul D. Hanna, Oct 04 2023

Keywords

Comments

Related identities which hold formally for all Maclaurin series F(x):
(1) F(x) = (1/x) * Sum{n>=1} n^(n-1) * x^n * F(x)^n / (1 + n*x*F(x))^(n+1),
(2) F(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n * F(x)^n / (1 + (n+1)*x*F(x))^(n+1),
(3) F(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n * F(x)^n / (1 + (n+2)*x*F(x))^(n+1),
(4) F(x) = (4/x) * Sum{n>=1} n*(n+3)^(n-2) * x^n * F(x)^n / (1 + (n+3)*x*F(x))^(n+1),
(5) F(x) = (k/x) * Sum{n>=1} n*(n+k-1)^(n-2) * x^n * F(x)^n / (1 + (n+k-1)*x*F(x))^(n+1) for all fixed nonzero k.

Examples

			G.f.: A(x) = 1 + x + 9*x^2 + 73*x^3 + 625*x^4 + 5681*x^5 + 53945*x^6 + 528697*x^7 + 5307489*x^8 + 54298849*x^9 + 564079337*x^10 + ...
where A(x) satisfies A(x) = 1 + x*A(x)/(1 - 4*x*A(x))^2
also
A(x) = 1 + 1^0*x*A(x)/(1 + (-3)*x*A(x))^2 + 2^1*x^2*A(x)^2/(1 + (-2)*x*A(x))^3 + 3^2*x^3*A(x)^3/(1 + (-1)*x*A(x))^4 + 4^3*x^4*A(x)^4/(1 + 0*x*A(x))^5 + 5^4*x^5*A(x)^5/(1 + 1*x*A(x))^6 + 6^5*x^6*A(x)^6/(1 + 2*x*A(x))^7 + ...
and
A(x) = 1 + 5*1*5^(-1)*x*A(x)/(1 + 1*x*A(x))^2 + 5*2*6^0*x^2*A(x)^2/(1 + 2*x*A(x))^3 + 5*3*7^1*x^3*A(x)^3/(1 + 3*x*A(x))^4 + 5*4*8^2*x^4*A(x)^4/(1 + 4*x*A(x))^5 + 5*5*9^3*x^5*A(x)^5/(1 + 5*x*A(x))^6 + ...

Links

Crossrefs

Cf. A365770, A109081, A365772, A365773, A365775.
Cf. A366234 (dual).

Programs

  • PARI
    {a(n) = sum(k=0, n, binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 4^k)}
for(n=0,30, print1(a(n),", "))
  • PARI
    {a(n) = my(A = (1/x) * serreverse( x/(1 + x/(1 - 4*x +O(x^(n+2)) )^2) ) ); polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))

Formula

a(n) = Sum{k=0..n} binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 4^k.
Let A(x)^m = Sum_{n>=0} a(n,m) * x^n then a(n,m) = Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * binomial(2*n-k-1, k) * 4^k.
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*A(x)/(1 - 4*x*A(x))^2.
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x/(1 - 4*x)^2) ).
(3) A( x/(1 + x/(1 - 4*x)^2) ) = 1 + x/(1 - 4*x)^2.
(4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n * A(x)^n / (1 + (n+m-4)*x*A(x))^(n+1) for all fixed nonnegative m.
(4.a) A(x) = 1 + Sum{n>=1} n^(n-1) * x^n * A(x)^n / (1 + (n-4)*x*A(x))^(n+1).
(4.b) A(x) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n * A(x)^n / (1 + (n-3)*x*A(x))^(n+1).
(4.c) A(x) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n * A(x)^n / (1 + (n-2)*x*A(x))^(n+1).
(4.d) A(x) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n * A(x)^n / (1 + (n-1)*x*A(x))^(n+1).
(4.e) A(x) = 1 + 5 * Sum{n>=1} n*(n+4)^(n-2) * x^n * A(x)^n / (1 + n*x*A(x))^(n+1).

A365775 Expansion of g.f. A(x) satisfying A(x) = 1 + x*A(x)/(1 - 5*x*A(x))^2.

Original entry on oeis.org

1, 1, 11, 106, 1061, 11226, 124026, 1414211, 16515981, 196551736, 2375042076, 29062573926, 359407971786, 4484868410231, 56399986492661, 714067825064426, 9094408567049701, 116436367409647736, 1497734068943432856, 19346547929074098836, 250851388061224003276
Offset: 0

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Author

Paul D. Hanna, Oct 04 2023

Keywords

Comments

Related identities which hold formally for all Maclaurin series F(x):
(1) F(x) = (1/x) * Sum{n>=1} n^(n-1) * x^n * F(x)^n / (1 + n*x*F(x))^(n+1),
(2) F(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n * F(x)^n / (1 + (n+1)*x*F(x))^(n+1),
(3) F(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n * F(x)^n / (1 + (n+2)*x*F(x))^(n+1),
(4) F(x) = (4/x) * Sum{n>=1} n*(n+3)^(n-2) * x^n * F(x)^n / (1 + (n+3)*x*F(x))^(n+1),
(5) F(x) = (k/x) * Sum{n>=1} n*(n+k-1)^(n-2) * x^n * F(x)^n / (1 + (n+k-1)*x*F(x))^(n+1) for all fixed nonzero k.

Examples

			G.f.: A(x) = 1 + x + 11*x^2 + 106*x^3 + 1061*x^4 + 11226*x^5 + 124026*x^6 + 1414211*x^7 + 16515981*x^8 + 196551736*x^9 + 2375042076*x^10 + ...
where A(x) satisfies A(x) = 1 + x*A(x)/(1 - 5*x*A(x))^2
also
A(x) = 1 + 1^0*x*A(x)/(1 + (-4)*x*A(x))^2 + 2^1*x^2*A(x)^2/(1 + (-3)*x*A(x))^3 + 3^2*x^3*A(x)^3/(1 + (-2)*x*A(x))^4 + 4^3*x^4*A(x)^4/(1 + (-1)*x*A(x))^5 + 5^4*x^5*A(x)^5/(1 + 0*x*A(x))^6 + 6^5*x^6*A(x)^6/(1 + 1*x*A(x))^7 + ...
and
A(x) = 1 + 6*1*6^(-1)*x*A(x)/(1 + 1*x*A(x))^2 + 6*2*7^0*x^2*A(x)^2/(1 + 2*x*A(x))^3 + 6*3*8^1*x^3*A(x)^3/(1 + 3*x*A(x))^4 + 6*4*9^2*x^4*A(x)^4/(1 + 4*x*A(x))^5 + 6*5*10^3*x^5*A(x)^5/(1 + 5*x*A(x))^6 + ...

Links

Crossrefs

Cf. A365770, A109081, A365772, A365773, A365774.
Cf. A366235 (dual).

Programs

  • PARI
    {a(n) = sum(k=0, n, binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 5^k)}
for(n=0,30, print1(a(n),", "))
  • PARI
    {a(n) = my(A = (1/x) * serreverse( x/(1 + x/(1 - 5*x +O(x^(n+2)) )^2) ) ); polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))

Formula

a(n) = Sum{k=0..n} binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 5^k.
Let A(x)^m = Sum_{n>=0} a(n,m) * x^n then a(n,m) = Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * binomial(2*n-k-1, k) * 5^k.
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*A(x)/(1 - 5*x*A(x))^2.
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x/(1 - 5*x)^2) ).
(3) A( x/(1 + x/(1 - 5*x)^2) ) = 1 + x/(1 - 5*x)^2.
(4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n * A(x)^n / (1 + (n+m-5)*x*A(x))^(n+1) for all fixed nonnegative m.
(4.a) A(x) = 1 + Sum{n>=1} n^(n-1) * x^n * A(x)^n / (1 + (n-5)*x*A(x))^(n+1).
(4.b) A(x) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n * A(x)^n / (1 + (n-4)*x*A(x))^(n+1).
(4.c) A(x) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n * A(x)^n / (1 + (n-3)*x*A(x))^(n+1).
(4.d) A(x) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n * A(x)^n / (1 + (n-2)*x*A(x))^(n+1).
(4.e) A(x) = 1 + 5 * Sum{n>=1} n*(n+4)^(n-2) * x^n * A(x)^n / (1 + (n-1)*x*A(x))^(n+1).
(4.f) A(x) = 1 + 6 * Sum{n>=1} n*(n+5)^(n-2) * x^n * A(x)^n / (1 + n*x*A(x))^(n+1).
a(n) ~ sqrt(3) * 5^(2*n) * (19^(3/2 + n) / (2*sqrt((113 + (28*(47225 + 1083*sqrt(1905))^(1/3))/5^(2/3) - 2632/(5*(47225 + 1083*sqrt(1905)))^(1/3))*Pi) * n^(3/2) * (68 + (2*(-1496331 + 60325*sqrt(1905)))^(1/3)/3^(2/3) - 9214*2^(2/3)/(3*(-1496331 + 60325*sqrt(1905)))^(1/3))^(n + 1/2))). - Vaclav Kotesovec, Oct 06 2023
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