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.

A303056 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n - A(x))^n.

Original entry on oeis.org

1, 1, 1, 8, 89, 1326, 24247, 521764, 12867985, 357229785, 11017306489, 373675921093, 13825260663882, 554216064798423, 23934356706763264, 1108017262467214486, 54747529760516714323, 2876096694574711401525, 160092696678371426933342, 9413031424290635395882462, 583000844360279565483710624
Offset: 0

Views

Author

Paul D. Hanna, Apr 19 2018

Keywords

Comments

More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - r*p*q^n)^(n+k),
for any fixed integer k; here, k = 1 with r = 1, p = -A(x), q = (1+x). - Paul D. Hanna, Jun 22 2019

Examples

			G.f.: A(x) = 1 + x + x^2 + 8*x^3 + 89*x^4 + 1326*x^5 + 24247*x^6 + 521764*x^7 + 12867985*x^8 + 357229785*x^9 + 11017306489*x^10 + ...
such that
1 = 1  +  ((1+x) - A(x))  +  ((1+x)^2 - A(x))^2  +  ((1+x)^3 - A(x))^3  +  ((1+x)^4 - A(x))^4  +  ((1+x)^5 - A(x))^5  +  ((1+x)^6 - A(x))^6  +  ((1+x)^7 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1+x)/(1 + (1+x)*A(x))^2  +  (1+x)^4/(1 + (1+x)^2*A(x))^3  +  (1+x)^9/(1 + (1+x)^3*A(x))^4  +  (1+x)^16/(1 + (1+x)^4*A(x))^5  +  (1+x)^25/(1 + (1+x)^5*A(x))^6  +  (1+x)^36/(1 + (1+x)^6*A(x))^7 + ...
RELATED SERIES.
log(A(x)) = x + x^2/2 + 22*x^3/3 + 325*x^4/4 + 6186*x^5/5 + 137380*x^6/6 + 3478651*x^7/7 + 98674253*x^8/8 + 3096911434*x^9/9 + ...
PARTICULAR VALUES.
Although the power series A(x) diverges at x = -1/2, it may be evaluated formally.
Let t = A(-1/2) = 0.545218973635949431234950245034944106957612798888179456724264...
then t satisfies
(1) 1 = Sum_{n>=0} ( 1/2^n - t )^n.
(2) 1 = Sum_{n>=0} 2^n / ( 2^n + t )^(n+1).
Also,
A(r) = 1/2 at r = -0.54683649902292991492196620520872286547799291909992048564578...
where
(1) 1 = Sum_{n>=0} ( (1+r)^n - 1/2 )^n.
(2) 1 = Sum_{n>=0} (1+r)^(-n) / ( 1/(1+r)^n + 1/2 )^(n+1).

Links

Crossrefs

Cf. A304642, A304639, A303926.
Cf. A321602, A321603, A321604, A321605.
Cf. A326282, A326283, A326284.
Cf. A337755, A337756, A337757.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = Vec( sum(m=0,#A, ((1+x)^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)^n - A(x))^n.
(2) 1 = Sum_{n>=0} (1+x)^(n^2) / (1 + (1+x)^n*A(x))^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.1610886538654... and c = 0.11739505492506... - Vaclav Kotesovec, Sep 26 2020

A321602 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(2*n) - A(x))^n.

Original entry on oeis.org

1, 2, 5, 68, 1521, 45328, 1660032, 71548008, 3533826841, 196432984748, 12128132342482, 823366216285428, 60966207548525287, 4890600994792550264, 422601696583826709492, 39142599000082019249968, 3869325702147169825040193, 406650337650126697706078146, 45281361448272561712508294157, 5325916931170845646048163850556, 659842223101960470758187538118437
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2018

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 68*x^3 + 1521*x^4 + 45328*x^5 + 1660032*x^6 + 71548008*x^7 + 3533826841*x^8 + 196432984748*x^9 + 12128132342482*x^10 + ...
such that
1 = 1  +  ((1+x)^2 - A(x))  +  ((1+x)^4 - A(x))^2  +  ((1+x)^6 - A(x))^3  +  ((1+x)^8 - A(x))^4  +  ((1+x)^10 - A(x))^5  +  ((1+x)^12 - A(x))^6  +  ((1+x)^14 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1+x)^2/(1 + (1+x)^2*A(x))^2  +  (1+x)^8/(1 + (1+x)^4*A(x))^3  +  (1+x)^18/(1 + (1+x)^6*A(x))^4  +  (1+x)^32/(1 + (1+x)^8*A(x))^5  +  (1+x)^50/(1 + (1+x)^10*A(x))^6  + ...
RELATED SERIES.
The logarithmic derivative of the g.f. begins
A'(x)/A(x) = 2 + 6*x + 182*x^2 + 5554*x^3 + 211172*x^4 + 9397920*x^5 + 476737830*x^6 + 27086036234*x^7 + 1702330030676*x^8 + ...
the coefficients of which are all even:
(1/2) * A'(x)/A(x) = 1 + 3*x + 91*x^2 + 2777*x^3 + 105586*x^4 + 4698960*x^5 + 238368915*x^6 + 13543018117*x^7 + 851165015338*x^8 + ...

Links

Crossrefs

Cf. A303056, A321603, A321604, A321605.
Cf. A326262.

Programs

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

Formula

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

A321603 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(3*n) - A(x))^n.

Original entry on oeis.org

1, 3, 12, 235, 7872, 351924, 19340668, 1250971416, 92720438955, 7733929764167, 716488771114410, 72981787493017014, 8107675760704948748, 975749719762368998037, 126491959992115408069503, 17576241581408197850363955, 2606439876885873198662077692, 410925212330248782377865281826, 68641203626673300062880912740755, 12110976733338358608040713750036252
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2018

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 12*x^2 + 235*x^3 + 7872*x^4 + 351924*x^5 + 19340668*x^6 + 1250971416*x^7 + 92720438955*x^8 + 7733929764167*x^9 + ...
such that
1 = 1  +  ((1+x)^3 - A(x))  +  ((1+x)^6 - A(x))^2  +  ((1+x)^9 - A(x))^3  +  ((1+x)^12 - A(x))^4  +  ((1+x)^15 - A(x))^5  +  ((1+x)^18 - A(x))^6  +  ((1+x)^21 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1+x)^3/(1 + (1+x)^3*A(x))^2  +  (1+x)^12/(1 + (1+x)^6*A(x))^3  +  (1+x)^27/(1 + (1+x)^9*A(x))^4  +  (1+x)^48/(1 + (1+x)^12*A(x))^5  +  (1+x)^75/(1 + (1+x)^15*A(x))^6  + ...
RELATED SERIES.
The logarithmic derivative of the g.f. begins
A'(x)/A(x) = 3 + 15*x + 624*x^2 + 28731*x^3 + 1638798*x^4 + 109462350*x^5 + 8333782509*x^6 + 710574703107*x^7 + 67015908514587*x^8 + ...
the coefficients of which are all divisible by 3:
(1/3) * A'(x)/A(x) = 1 + 5*x + 208*x^2 + 9577*x^3 + 546266*x^4 + 36487450*x^5 + 2777927503*x^6 + 236858234369*x^7 + 22338636171529*x^8 + ...

Links

Crossrefs

Cf. A303056, A321602, A321604, A321605.
Cf. A326263.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(3*m) - Ser(A))^m ) )[#A] );H=A; A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^(3*n) - A(x))^n.
(2) 1 = Sum_{n>=0} (1+x)^(3*n^2) / (1 + (1+x)^(3*n)*A(x))^(n+1).

A321604 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(4*n) - A(x))^n.

Original entry on oeis.org

1, 4, 22, 564, 25157, 1499576, 109904860, 9480509576, 937113401201, 104240673195936, 12878161899791760, 1749261564410844864, 259132759251207789056, 41585481940418457992816, 7188476201158569394613976, 1331880173688346226092103696, 263358773243148578509342224153, 55363099822436514905885084770968, 12330972024423209530808891225876436, 2900976547500300324930009436969260936
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2018

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 22*x^2 + 564*x^3 + 25157*x^4 + 1499576*x^5 + 109904860*x^6 + 9480509576*x^7 + 937113401201*x^8 + 104240673195936*x^9 + ...
such that
1 = 1  +  ((1+x)^4 - A(x))  +  ((1+x)^8 - A(x))^2  +  ((1+x)^12 - A(x))^3  +  ((1+x)^16 - A(x))^4  +  ((1+x)^20 - A(x))^5  +  ((1+x)^24 - A(x))^6  +  ((1+x)^28 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1+x)^4/(1 + (1+x)^4*A(x))^2  +  (1+x)^16/(1 + (1+x)^8*A(x))^3  +  (1+x)^36/(1 + (1+x)^12*A(x))^4  +  (1+x)^64/(1 + (1+x)^16*A(x))^5  +  (1+x)^100/(1 + (1+x)^20*A(x))^6  + ...
RELATED SERIES.
The logarithmic derivative of the g.f. begins
A'(x)/A(x) = 4 + 28*x + 1492*x^2 + 91788*x^3 + 6981484*x^4 + 621939700*x^5 + 63151305340*x^6 + 7181135905380*x^7 + 903210250234696*x^8 + ...
the coefficients of which are all divisible by 4:
(1/4) * A'(x)/A(x) = 1 + 7*x + 373*x^2 + 22947*x^3 + 1745371*x^4 + 155484925*x^5 + 15787826335*x^6 + 1795283976345*x^7 + 225802562558674*x^8 + ...

Links

Crossrefs

Cf. A303056, A321602, A321603, A321605.
Cf. A326264.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(4*m) - Ser(A))^m ) )[#A] );H=A; A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^(4*n) - A(x))^n.
(2) 1 = Sum_{n>=0} (1+x)^(4*n^2) / (1 + (1+x)^(4*n)*A(x))^(n+1).

A326265 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1/(1-x)^(5*n) - A(x) )^n.

Original entry on oeis.org

1, 5, 40, 1185, 65270, 4861126, 445776670, 48124064710, 5952881626790, 828544320379330, 128058593506875627, 21758230559633783765, 4031357498037096661170, 809070343591564791211705, 174888309616496370413590235, 40517215307075701804767255261, 10017278630199891781122121185615, 2632883558256463087445119555912870, 733167697272377998186394054589647855, 215641985221691590110546294934099963285
Offset: 0

Views

Author

Paul D. Hanna, Jun 20 2019

Keywords

Examples

			G.f.: A(x) = 1 + 5*x + 40*x^2 + 1185*x^3 + 65270*x^4 + 4861126*x^5 + 445776670*x^6 + 48124064710*x^7 + 5952881626790*x^8 + 828544320379330*x^9 + 128058593506875627*x^10 + ...
such that
1 = 1  +  (1/(1-x)^5 - A(x))  +  (1/(1-x)^10 - A(x))^2  +  (1/(1-x)^15 - A(x))^3  +  (1/(1-x)^20 - A(x))^4  +  (1/(1-x)^25 - A(x))^5  +  (1/(1-x)^30 - A(x))^6  +  (1/(1-x)^35 - A(x))^7  + ...
Also,
1 = 1/(1 + A(x))  +  (1-x)^5/((1-x)^5 + A(x))^2  +  (1-x)^10/((1-x)^10 + A(x))^3  +  (1-x)^15/((1-x)^15  +  A(x))^4 + (1-x)^20/((1-x)^20 + A(x))^5  +  (1-x)^25/((1-x)^25 + A(x))^6  +  (1-x)^30/((1-x)^30 + A(x))^7 + ...

Crossrefs

Cf. A304639, A326262, A326263, A326264.
Cf. A321605.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1-x)^(-5*m) - Ser(A))^m ) )[#A] ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/(1-x)^(5*n) - A(x) )^n.
(2) 1 = Sum_{n>=0} ( 1 - (1-x)^(5*n)*A(x) )^n / (1-x)^(5*n^2).
(3) 1 = Sum_{n>=0} (1-x)^(5*n) / ( (1-x)^(5*n) + A(x) )^(n+1).
Showing 1-5 of 5 results.

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