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 1-4 of 4 results.

A122400 Number of square (0,1)-matrices without zero rows and with exactly n entries equal to 1.

Original entry on oeis.org

1, 1, 4, 31, 338, 4769, 82467, 1687989, 39905269, 1069863695, 32071995198, 1062991989013, 38596477083550, 1523554760656205, 64961391010251904, 2975343608212835855, 145687881987604377815, 7594435556630244257213
Offset: 0

Views

Author

Vladeta Jovovic, Aug 31 2006

Keywords

Links

Crossrefs

Cf. A104602, A220353, A301581, A301582, A301583, A301584.

Programs

  • Maple
    A122399 := proc(n) option remember ; add( combinat[stirling2](n,k)*k^n*k!,k=0..n) ; end: A122400 := proc(n) option remember ; add( combinat[stirling1](n,k)*A122399(k),k=0..n)/n! ; end: for n from 0 to 30 do printf("%d, ",A122400(n)) ; od ; # R. J. Mathar, May 18 2007
  • Mathematica
    max = 17; CoefficientList[ Series[ 1 + Sum[ ((1 + x)^n - 1)^n, {n, 1, max}], {x, 0, max}], x] (* Jean-François Alcover, Mar 26 2013, after Vladeta Jovovic *)

Formula

a(n) = (1/n!)* Sum_{k=0..n} Stirling1(n,k)*A122399(k).
G.f.: Sum_{n>=0} ((1+x)^n - 1)^n. - Vladeta Jovovic, Sep 03 2006
G.f.: Sum_{n>=0} (1+x)^(n^2) / (1 + (1+x)^n)^(n+1). - Paul D. Hanna, Mar 23 2018
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.2796968489586733500739737080739303725411427162653658... . - Vaclav Kotesovec, May 07 2014

Extensions

More terms from R. J. Mathar, May 18 2007

A317855 Decimal expansion of a constant related to the asymptotics of A122400.

Original entry on oeis.org

3, 1, 6, 1, 0, 8, 8, 6, 5, 3, 8, 6, 5, 4, 2, 8, 8, 1, 3, 8, 3, 0, 1, 7, 2, 2, 0, 2, 5, 8, 8, 1, 3, 2, 4, 9, 1, 7, 2, 6, 3, 8, 2, 7, 7, 4, 1, 8, 8, 5, 5, 6, 3, 4, 1, 6, 2, 7, 2, 7, 8, 2, 0, 7, 5, 3, 7, 6, 9, 7, 0, 5, 9, 2, 1, 9, 3, 0, 4, 6, 1, 1, 2, 1, 9, 7, 5, 7, 4, 6, 8, 5, 4, 9, 7, 8, 4, 5, 9, 3, 2, 4, 2, 2, 7
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 09 2018

Keywords

Examples

			3.161088653865428813830172202588132491726382774188556341627278...

Crossrefs

Cf. A121886, A122399, A122400, A122418, A122419, A122420, A227619, A232192, A243802, A244585, A248798, A317340, A326010.
Cf. A301584, A301585, A301586, A303056, A303057, A303654.

Programs

  • Mathematica
    r = r /. FindRoot[E^(1/r)/r + (1 + E^(1/r)) * ProductLog[-E^(-1/r)/r] == 0, {r, 3/4}, WorkingPrecision -> 120]; RealDigits[(1 + Exp[1/r])*r^2][[1]]
  • PARI
    r=solve(r=.8,1,exp(1/r)/r + (1+exp(1/r))*lambertw(-exp(-1/r)/r))
(1+exp(1/r))*r^2 \\ Charles R Greathouse IV, Jun 15 2021

Formula

Equals (1+exp(1/r))*r^2, where r = 0.873702433239668330496568304720719298213992... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0.

A301585 G.f.: Sum_{n>=0} ((1+x)^(3*n) - 1)^n.

Original entry on oeis.org

1, 3, 39, 910, 29949, 1271751, 66116065, 4066082856, 288701376912, 23240635243591, 2091554595246705, 208085119389952134, 22676957610808295192, 2686515300821612112411, 343760257348413122290260, 47248346582443326267328400, 6942339982115290619799947901, 1085919469129099832397573088863, 180160797497273341662653292624309, 31598815412054398239059538582525618
Offset: 0

Views

Author

Paul D. Hanna, Mar 24 2018

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 39*x^2 + 910*x^3 + 29949*x^4 + 1271751*x^5 + 66116065*x^6 + 4066082856*x^7 + 288701376912*x^8 + ...
such that
A(x) = 1 + ((1+x)^3-1) + ((1+x)^6-1)^2 + ((1+x)^9-1)^3 + ((1+x)^12-1)^4 + ((1+x)^15-1)^5 + ((1+x)^18-1)^6 + ((1+x)^21-1)^7 + ...
Also,
A(x) = 1/2 + (1+x)^3/(1 + (1+x)^3)^2 + (1+x)^12/(1 + (1+x)^6)^3 + (1+x)^27/(1 + (1+x)^9)^4 + (1+x)^48/(1 + (1+x)^12)^5 + (1+x)^75/(1 + (1+x)^15)^6 + ...

Links

Crossrefs

Cf. A122400, A301584, A301586.

Programs

  • PARI
    {a(n) = my(A,o=x*O(x^n)); A = sum(m=0,n, ((1+x +o)^(3*m) - 1)^m ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} (1+x)^(3*n^2) /(1 + (1+x)^(3*n))^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = 3*A317855 = 9.4832659615962864414905166077643974751791483225656690248818346226130911776579... and c = 0.3108017465925995208675813879173750641359609... - Vaclav Kotesovec, Aug 09 2018

A301586 G.f.: Sum_{n>=0} ((1+x)^(4*n) - 1)^n.

Original entry on oeis.org

1, 4, 70, 2180, 95729, 5422192, 375951144, 30833206304, 2919367902648, 313380517364324, 37606931999739230, 4988933437333555060, 724960700435104219679, 114519163835687116024256, 19538926882901715534673728, 3580844611314789257667535968, 701546780854024941112271649610, 146318317830136401429653726419700, 32367591848747955557013839920695374, 7569528177000020896435962191564396740
Offset: 0

Views

Author

Paul D. Hanna, Mar 24 2018

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 70*x^2 + 2180*x^3 + 95729*x^4 + 5422192*x^5 + 375951144*x^6 + 30833206304*x^7 + ...
such that
A(x) = 1 + ((1+x)^4-1) + ((1+x)^8-1)^2 + ((1+x)^12-1)^3 + ((1+x)^16-1)^4 + ((1+x)^20-1)^5 + ((1+x)^24-1)^6 + ((1+x)^28-1)^7 + ...
Also,
A(x) = 1/2 + (1+x)^4/(1 + (1+x)^4)^2 + (1+x)^16/(1 + (1+x)^8)^3 + (1+x)^36/(1 + (1+x)^12)^4 + (1+x)^64/(1 + (1+x)^16)^5 + (1+x)^100/(1 + (1+x)^20)^6 + ...

Links

Crossrefs

Cf. A122400, A301584, A301585.

Programs

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

Formula

G.f.: Sum_{n>=0} (1+x)^(4*n^2) /(1 + (1+x)^(4*n))^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = 4*A317855 = 12.64435461546171525532068881035252996690553109675422536650911283015078823687... and c = 0.31492557816516652573983016205911709623053... - Vaclav Kotesovec, Aug 09 2018
Showing 1-4 of 4 results.

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