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 1-3 of 3 results.

A303926 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1 + x*A(x))^n - A(x) )^n.

Original entry on oeis.org

1, 1, 2, 12, 130, 1912, 34715, 743217, 18255118, 505070221, 15532353184, 525533183871, 19403298048040, 776437898905606, 33479679336072541, 1547841068340501230, 76390272348430998076, 4008960603544297652028, 222949077434693015546579, 13098226217965693342007714, 810657425687536689904281842
Offset: 0

Views

Author

Paul D. Hanna, May 03 2018

Keywords

Comments

Compare to: 1 = Sum_{n>=0} ( (1 + x*G(x))^k - G(x) )^n holds trivially for fixed k>0 when G(x) = (1 + x*G(x))^k ; this sequence explores the case when k varies with n.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 130*x^4 + 1912*x^5 + 34715*x^6 + 743217*x^7 + 18255118*x^8 + 505070221*x^9 + 15532353184*x^10 + ...
such that
1 = 1 + ((1 + x*A(x)) - A(x)) + ((1 + x*A(x))^2 - A(x))^2 + ((1 + x*A(x))^3 - A(x))^3 + ((1 + x*A(x))^4 - A(x))^4 + ((1 + x*A(x))^5 - A(x))^5 + ...

Crossrefs

Cf. A303927, A303928, A303923, A303056.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = Vec( sum(m=0,#A, ( (1 + x*Ser(A))^m - Ser(A))^m ) )[#A] ); A[n+1]}
for(n=0,30, print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( (1 + x*A(x))^n - A(x) )^n.
(2) 1 = Sum_{n>=0} (1 + x*A(x))^(n^2) / (1 + A(x)*(1 + x*A(x))^n)^(n+1). - Paul D. Hanna, Dec 06 2018
G.f.: x/Series_Reversion( x*F(x) ) such that 1 = Sum_{n>=0} ((1 + x*F(x)^2)^n - F(x))^n, where F(x) is the g.f. of A303927.
G.f.: sqrt( x/Series_Reversion( x*G(x)^2 ) ) such that 1 = Sum_{n>=0} ((1 + x*G(x)^3)^n - G(x))^n, where G(x) is the g.f. of A303928.

A303924 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1 + x*A(x)^(n+1) - A(x) )^n.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 204, 891, 4266, 22092, 122358, 718282, 4438154, 28711805, 193700970, 1358588449, 9883071724, 74423630202, 579231718432, 4652864427983, 38528749877802, 328519744186940, 2881366257269722, 25969840412367362, 240307819488203558, 2280902112035109237, 22187847195528993904, 221024332987155498348
Offset: 0

Views

Author

Paul D. Hanna, May 03 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 204*x^6 + 891*x^7 + 4266*x^8 + 22092*x^9 + 122358*x^10 + 718282*x^11 + 4438154*x^12 + ...
such that
1 = 1 + (1 + x*A(x)^2 - A(x)) + (1 + x*A(x)^3 - A(x))^2 + (1 + x*A(x)^4 - A(x))^3 + (1 + x*A(x)^5 - A(x))^4 + (1 + x*A(x)^6 - A(x))^5 + ...

Links

Crossrefs

Cf. A303923, A303925.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = Vec( sum(m=0,#A, ( 1 + x*Ser(A)^(m+1) - Ser(A))^m ) )[#A] ); A[n+1]}
for(n=0,30, print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1 + x*A(x)^(n+1) - A(x) )^n.
(2) 1 = Sum_{n>=0} x^n * A(x)^(n*(n+1)) / (1 + (A(x)-1)*A(x)^n)^(n+1). - Paul D. Hanna, Dec 11 2018
G.f.: 1/x*Series_Reversion( x/F(x) ) such that 1 = Sum_{n>=0} ((1 + x*F(x))^n - F(x))^n, where F(x) is the g.f. of A303923.
G.f.: x/Series_Reversion( x*G(x) ) such that 1 = Sum_{n>=0} ((1 + x*G(x))^(n+2) - G(x))^n, where G(x) is the g.f. of A303925.

A303925 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1 + x*A(x)^(n+2) - A(x) )^n.

Original entry on oeis.org

1, 1, 3, 12, 56, 288, 1587, 9222, 55957, 352267, 2290842, 15343839, 105634437, 746478622, 5409932286, 40189454704, 305972524737, 2387238374532, 19090018863000, 156496468777604, 1315509548959765, 11341506519584442, 100300906407392783, 909967403153604712, 8468614126450656268, 80832677102193209308, 791071858022525348235
Offset: 0

Views

Author

Paul D. Hanna, May 03 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 56*x^4 + 288*x^5 + 1587*x^6 + 9222*x^7 + 55957*x^8 + 352267*x^9 + 2290842*x^10 + 15343839*x^11 + ...
such that
1 = 1 + (1 + x*A(x)^3 - A(x)) + (1 + x*A(x)^4 - A(x))^2 + (1 + x*A(x)^5 - A(x))^3 + (1 + x*A(x)^6 - A(x))^4 + (1 + x*A(x)^7 - A(x))^5 + ...

Links

Crossrefs

Cf. A303923, A303924.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = Vec( sum(m=0,#A, ( 1 + x*Ser(A)^(m+2) - Ser(A))^m ) )[#A] ); A[n+1]}
for(n=0,30, print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1 + x*A(x)^(n+2) - A(x) )^n.
(2) 1 = Sum_{n>=0} x^n * A(x)^(n*(n+2)) / (1 + (A(x)-1)*A(x)^n)^(n+1). - Paul D. Hanna, Dec 11 2018
G.f.: 1/x*Series_Reversion( x/F(x) ) such that 1 = Sum_{n>=0} ((1 + x*F(x))^(n+1) - F(x))^n, where F(x) is the g.f. of A303924.
G.f.: sqrt( 1/x*Series_Reversion( x/G(x)^2 ) ) such that 1 = Sum_{n>=0} ((1 + x*G(x))^n - G(x))^n, where G(x) is the g.f. of A303923.
Showing 1-3 of 3 results.

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