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 1-6 of 6 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

A338040 E.g.f.: Sum_{j>=0} 4^j * (exp(j*x) - 1)^j.

Original entry on oeis.org

1, 4, 132, 11140, 1763076, 449262724, 168055179012, 86720706877060, 59029852191779076, 51241585497612147844, 55245853646893977682692, 72423868722672448652558980, 113447698393867318106045295876, 209271794145089904620369489016964
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 08 2020

Keywords

Comments

In general, if k > 0 and e.g.f.: Sum_{j>=0} k^j * (exp(j*x) - 1)^j, then a(n) ~ c * (1 + k*exp(1/r))^n * r^(2*n) * n!^2 / sqrt(n), where r is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/k and c is a constant (dependent only on k).
Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [4, 6, 3, 0, 1, 0, 4, 6, 3, 0, 1, 0, 4, 6, 3, 0, 1, 0, ...], with an apparent period of 6. - Peter Bala, May 31 2022

Links

Crossrefs

Cf. A122399, A195005, A195263, A195415, A220181, A221077, A221078, A224899, A245322.
Cf. A122400, A301581, A301582, A301583.

Programs

  • Mathematica
    Flatten[{1, Table[Sum[4^j * j^n * j! * StirlingS2[n, j], {j, 0, n}], {n, 1, 20}]}]
nmax = 20; CoefficientList[Series[1 + Sum[4^j*(Exp[j*x] - 1)^j, {j, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
  • PARI
    {a(n)=local(X=x+x*O(x^n)); n!*polcoeff(sum(m=0, n, 4^m*(exp(m*X)-1)^m), n)}

Formula

a(n) = Sum_{j=0..n} 4^j * j^n * j! * Stirling2(n,j).
a(n) ~ c * (1 + 4*exp(1/r))^n * r^(2*n) * n!^2 / sqrt(n), where r = 0.95894043087329419322124137165060249611787608513866855417024... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/4 and c = 0.37483929689722634406486945426531890297038414869116425498643733178324...

A301581 G.f.: Sum_{n>=0} 2^n * ((1+x)^n - 1)^n.

Original entry on oeis.org

1, 2, 16, 232, 4748, 125440, 4058312, 155336672, 6864980968, 343995674280, 19270975801600, 1193481831243584, 80966964261458368, 5971270693661978816, 475655179279901897536, 40699219246551726635840, 3722813577249648564213392, 362519587815189751405383520, 37442485808471509306691295808, 4088344078912544484116541775616, 470550859964811044524886252649760
Offset: 0

Views

Author

Paul D. Hanna, Mar 24 2018

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 16*x^2 + 232*x^3 + 4748*x^4 + 125440*x^5 + 4058312*x^6 + 155336672*x^7 + 6864980968*x^8 + 343995674280*x^9 + ...
such that
A(x) = 1 + 2*((1+x)-1) + 4*((1+x)^2-1)^2 + 8*((1+x)^3-1)^3 + 16*((1+x)^4-1)^4 + 32*((1+x)^5-1)^5 + 64*((1+x)^6-1)^6 + 128*((1+x)^7-1)^7 + ...
Also,
A(x) = 1/3 + 2*(1+x)/(1 + 2*(1+x))^2 + 4*(1+x)^4/(1 + 2*(1+x)^2)^3 + 8*(1+x)^9/(1 + 2*(1+x)^3)^4 + 16*(1+x)^16/(1 + 2*(1+x)^4)^5 + 32*(1+x)^25/(1 + 2*(1+x)^5)^6 + ...

Links

Crossrefs

Cf. A122400, A195005, A301582, A301583, A301463.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[1 + Sum[2^j*((1 + x)^j - 1)^j, {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2020 *)
  • PARI
    {a(n) = my(A,o=x*O(x^n)); A = sum(m=0,n, 2^m * ((1+x +o)^m - 1)^m ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} 2^n * (1+x)^(n^2) /(1 + 2*(1+x)^n)^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = (1 + 2*exp(1/r)) * r^2 = 5.90397654567002180048849478643459748789951071189960380570421209622277299181878..., where r = 0.925556278640887084941460444526398190071550948416... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/2 and c = 0.3270084259351451234078681822585067892123772641092056... - Vaclav Kotesovec, Aug 09 2018, updated Oct 08 2020

A301582 G.f.: Sum_{n>=0} 3^n * ((1+x)^n - 1)^n.

Original entry on oeis.org

1, 3, 36, 765, 22932, 886707, 41971041, 2349915543, 151893243711, 11131097539221, 911906584505874, 82586031357156975, 8192750710914222984, 883506535094875209327, 102907862475072248379060, 12875067336646598300376165, 1722014444866824121524712497, 245185575019136812676809863351, 37027348593726417935247243009495, 5911490521308027393188499233189367, 994821814352463817234026392636083551
Offset: 0

Views

Author

Paul D. Hanna, Mar 24 2018

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 36*x^2 + 765*x^3 + 22932*x^4 + 886707*x^5 + 41971041*x^6 + 2349915543*x^7 + 151893243711*x^8 + ...
such that
A(x) = 1 + 3*((1+x)-1) + 9*((1+x)^2-1)^2 + 27*((1+x)^3-1)^3 + 81*((1+x)^4-1)^4 + 243*((1+x)^5-1)^5 + 729*((1+x)^6-1)^6 + 2187*((1+x)^7-1)^7 + ...
Also,
A(x) = 1/4 + 3*(1+x)/(1 + 3*(1+x))^2 + 9*(1+x)^4/(1 + 3*(1+x)^2)^3 + 27*(1+x)^9/(1 + 3*(1+x)^3)^4 + 81*(1+x)^16/(1 + 3*(1+x)^4)^5 + 243*(1+x)^25/(1 + 3*(1+x)^5)^6 + ...

Links

Crossrefs

Cf. A122400, A195263, A301581, A301583, A301463.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[1 + Sum[3^j*((1 + x)^j - 1)^j, {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2020 *)
  • PARI
    {a(n) = my(A,o=x*O(x^n)); A = sum(m=0,n, 3^m * ((1+x +o)^m - 1)^m ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} 3^n * (1+x)^(n^2) /(1 + 3*(1+x)^n)^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = (1 + 3*exp(1/r)) * r^2 = 8.632012704198046828204904686098781240870113556702123911346365466059061495897353..., where r = 0.947093169766093813913446822751643203941993193936... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/3 and c = 0.34734097623709084937300542950550592394946492732014... - Vaclav Kotesovec, Aug 09 2018

A301463 G.f.: Sum_{n>=0} (2*(1+x)^n - 1)^n / 2^(n+1).

Original entry on oeis.org

1, 6, 134, 5102, 272694, 18758134, 1577807110, 156883546142, 18001728695894, 2341268080847014, 340346951612008454, 54686371000455538574, 9624103747115691611318, 1841049154379441320293142, 380367456989975381891133446, 84407842226680664984458744126, 20023121531700221583865582432854, 5056357801144690975957652265658438, 1354259474931265421064754160458035078, 383444904170987865090156939638756172846
Offset: 0

Views

Author

Paul D. Hanna, Mar 24 2018

Keywords

Examples

			G.f.: A(x) = 1 + 6*x + 134*x^2 + 5102*x^3 + 272694*x^4 + 18758134*x^5 + 1577807110*x^6 + 156883546142*x^7 + 18001728695894*x^8 + ...
such that
A(x) = 1/2  +  (2*(1+x) - 1)/2^2  +  (2*(1+x)^2 - 1)^2/2^3  +  (2*(1+x)^3 - 1)^3/2^4  +  (2*(1+x)^4 - 1)^4/2^5  +  (2*(1+x)^5 - 1)^5/2^6  + ...
Also,
A(x) = 1/3  +  2*(1+x)/(2 + (1+x))^2  +  2^2*(1+x)^4/(2 + (1+x)^2)^3  +  2^3*(1+x)^9/(2 + (1+x)^3)^4  +  2^4*(1+x)^16/(2 + (1+x)^4)^5  +  2^5*(1+x)^25/(2 + (1+x)^5)^6  +  2^6*(1+x)^36/(2 + (1+x)^6)^7  + ...

Links

Crossrefs

Cf. A122400, A301581, A301582, A301583.

Programs

  • Mathematica
    nmax = 20; Round[CoefficientList[Series[Sum[(2*(1 + x)^j - 1)^j/2^(j + 1), {j, 0, nmax^2}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 08 2020 *)

Formula

G.f.: Sum_{n>=0} 2^n * (1+x)^(n^2) / (2 + (1+x)^n)^(n+1).
G.f.: Sum_{n>=0} ((1+x)^n - 1/2)^n / 2.
a(n) ~ c * d^n * n! / sqrt(n), where d = 15.305828173910545025228605110120647795... and c = 0.4246982835243422293505427496472772728... - Vaclav Kotesovec, Aug 09 2018

Extensions

b-file confirmed by Vaclav Kotesovec, Oct 08 2020

A317799 G.f.: Sum_{n>=0} (4*(1+x)^n - 1)^n / 4^(n+1).

Original entry on oeis.org

1, 28, 2644, 418108, 92624756, 26388012380, 9189259388052, 3782063138596476, 1796136011427955636, 966755321167565129372, 581573928178258915024596, 386690499153558305585430460, 281600848152507182372274325492, 222904650325844057584524049181660, 190559248618061561787517993382005012
Offset: 0

Views

Author

Paul D. Hanna, Aug 14 2018

Keywords

Examples

			G.f.: A(x) = 1 + 28*x + 2644*x^2 + 418108*x^3 + 92624756*x^4 + 26388012380*x^5 + 9189259388052*x^6 + 3782063138596476*x^7 + 1796136011427955636*x^8 + ...
such that
A(x) = 1/4  +  (4*(1+x) - 1)/4^2  +  (4*(1+x)^2 - 1)^3/4^3  +  (4*(1+x)^3 - 1)^4/4^4  +  (4*(1+x)^4 - 1)^4/4^5  +  (4*(1+x)^5 - 1)^5/4^6  + ...
Also,
A(x) = 1/5  +  4*(1+x)/(4 + (1+x))^2  +  4^2*(1+x)^4/(4 + (1+x)^2)^4  +  4^3*(1+x)^9/(4 + (1+x)^3)^4  +  4^4*(1+x)^16/(4 + (1+x)^4)^5  +  4^5*(1+x)^25/(4 + (1+x)^5)^6  +  4^6*(1+x)^36/(4 + (1+x)^6)^7  + ...

Crossrefs

Cf. A122400, A301463, A317798, A301583.

Formula

G.f. satisfies:
(1) Sum_{n>=0} 4^n * (1+x)^(n^2) / (4 + (1+x)^n)^(n+1).
(2) Sum_{n>=0} ((1+x)^n - 1/4)^n / 4.
Showing 1-6 of 6 results.

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