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.

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A185040 O.g.f.: 1/(1-x) * Sum_{n>=0} 1/(1-(n+1)*x)^n * x^n/n! * exp(-x/(1-(n+1)*x)).

Original entry on oeis.org

1, 1, 2, 5, 15, 54, 220, 973, 4607, 23230, 124088, 698471, 4124961, 25474314, 164063103, 1099233251, 7645091839, 55085061358, 410472347944, 3158307976315, 25057152530411, 204717532709542, 1720324316575275, 14853374782672785, 131632834029683663, 1196258970969508760
Offset: 0

Views

Author

Paul D. Hanna, Dec 25 2012

Keywords

Comments

Compare g.f. to the curious identity:
1/(1+x^2) = Sum_{n>=0} (1-(n+1)*x)^n * x^n/n! * exp(-x*(1-(n+1)*x)).

Examples

			O.g.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 54*x^5 + 220*x^6 +...
where the o.g.f. equals the series:
A(x)*(1-x) = exp(-x/(1-x)) + x/(1-2*x)*exp(-x/(1-2*x)) + x^2/(1-3*x)^2/2!*exp(-x/(1-3*x)) + x^3/(1-4*x)^3/3!*exp(-x/(1-4*x)) + x^4/(1-5*x)^4/4!*exp(-x/(1-5*x)) + x^5/(1-6*x)^5/5!*exp(-x/(1-6*x)) + x^6/(1-7*x)^6/6!*exp(-x/(1-7*x)) +...
which simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218667.

Programs

  • Mathematica
    m = 26; Sum[1/(1 - (n+1)x)^n x^n Exp[-x/(1 - (n+1)x)]/n!, {n, 0, m}]/(1-x) + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Jan 27 2020 *)
  • PARI
    {a(n)=local(A=1+x, X=x+x*O(x^n)); A=1/(1-x)*sum(k=0, n, 1/(1-(k+1)*X)^k*x^k/k!*exp(-X/(1-(k+1)*X))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

Binomial transform of A218667.

A243942 O.g.f.: Sum_{n>=0} n^(2*n) * x^n / (1 - n*x)^n * exp( -n^2*x / (1 - n*x) ) / n!.

Original entry on oeis.org

1, 1, 8, 121, 2698, 79654, 2929238, 129004633, 6619919386, 387904397222, 25555935470016, 1869945551975658, 150459006927310348, 13203459856456213172, 1254972882696473807298, 128439184335788533011489, 14082139161229781077548346, 1646731810035799151750487814
Offset: 0

Views

Author

Paul D. Hanna, Aug 09 2014

Keywords

Examples

			O.g.f.: A(x) = 1 + x + 8*x^2 + 121*x^3 + 2698*x^4 + 79654*x^5 + 2929238*x^6 +...
where
A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^4*x^2/(1-2*x)^2*exp(-4*x/(1-2*x))/2! + 3^6*x^3/(1-3*x)^3*exp(-9*x/(1-3*x))/3! + 4^8*x^4/(1-4*x)^4*exp(-16*x/(1-4*x))/4! + 5^10*x^5/(1-5*x)^5*exp(-25*x/(1-5*x))/5! +...
simplifies to a power series in x with integer coefficients.
Illustrate the terms by:
a(1) = 1*1 = 1;
a(2) = 1*1 + 1*7 = 8;
a(3) = 1*1 + 2*15 + 1*90 = 121;
a(4) = 1*1 + 3*31 + 3*301 + 1*1701 = 2698;
a(5) = 1*1 + 4*63 + 6*966 + 4*7770 + 1*42525 = 79654; ...
where Stirling2(n+k,k) forms a rectangular table as follows:
1, 1,  1,   1,    1,     1,      1,      1, ...;
0, 1,  3,   6,   10,    15,     21,     28, ...;
0, 1,  7,  25,   65,   140,    266,    462, ...;
0, 1, 15,  90,  350,  1050,   2646,   5880, ...;
0, 1, 31, 301, 1701,  6951,  22827,  63987, ...;
0, 1, 63, 966, 7770, 42525, 179487, 627396, ...; ...

Crossrefs

Cf. A134055, A174845, A218667, A048993 (Stirling2).

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[n-1,k-1] * StirlingS2[n+k,k],{k,1,n}],{n,1,20}]}] (* Vaclav Kotesovec, Aug 11 2014 *)
  • PARI
    {a(n)=polcoeff(sum(k=0, n+1, (k^2*x)^k/(1-k*x)^k*exp(-k^2*x/(1-k*x+x*O(x^n)))/k!), n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    {Stirling2(n, k) = sum(j=0, k, (-1)^(k+j) * binomial(k, j) * j^n) / k!}
{a(n)=if(n==0, 1, sum(k=1, n, Stirling2(n+k, k) * binomial(n-1, k-1)))}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) = Sum_{k=1..n} C(n-1,k-1) * Stirling2(n+k,k) for n>0, a(0)=1.
a(n) = c * (r^2/((1-r)*(2*r-1)))^n * n^(n-1/2) / exp(n), where r = 0.859294411517830517100430385442711799997876163... is the root of the equation (1-r)*(1+r)/r^2 = -LambertW(-exp(-1-1/r)*(1+r)/r), and c = 0.4180257159270405799046057130547446708890452... . - Vaclav Kotesovec, Aug 11 2014

A245059 a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(n,k) * 2^(n-k) for n>0, a(0)=1, where S2(n,k) = A048993(n,k) are Stirling numbers of the 2nd kind.

Original entry on oeis.org

1, 1, 3, 17, 129, 1177, 12463, 149053, 1975473, 28628865, 449059179, 7562334793, 135837896769, 2588529249737, 52093016105575, 1102851978691749, 24480094135644513, 568066476383361793, 13745454515733689427, 346020796943921077057, 9043636093339718229697, 244954584886648170627641
Offset: 0

Views

Author

Paul D. Hanna, Jul 10 2014

Keywords

Examples

			O.g.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 129*x^4 + 1177*x^5 + 12463*x^6 +...
where
A(x) = 1 + x/(1-2*x)*exp(-x/(1-2*x)) + 2^2*x^2/(1-4*x)^2*exp(-2*x/(1-4*x))/2! + 3^3*x^3/(1-6*x)^3*exp(-3*x/(1-6*x))/3! + 4^4*x^4/(1-8*x)^4*exp(-4*x/(1-8*x))/4! +...
simplifies to a power series in x with integer coefficients.
Illustrate the definition of the terms by:
a(2) = 1*1*2 + 1*1 = 3;
a(3) = 1*1*2^2 + 2*3*2 + 1*1 = 17;
a(4) = 1*1*2^3 + 3*7*2^2 + 3*6*2 + 1*1 = 129;
a(5) = 1*1*2^4 + 4*15*2^3 + 6*25*2^2 + 4*10*2 + 1*1 = 1177;
a(6) = 1*1*2^5 + 5*31*2^4 + 10*90*2^3 + 10*65*2^2 + 5*15*2 + 1*1 = 12463; ...

Crossrefs

Cf. A134055, A245060, A218667, A218670.

Programs

  • PARI
    {a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1)*polcoeff(1/prod(i=0, k, 1-i*x +x*O(x^(n-k))), n-k)*2^(n-k)))}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff(sum(k=0, n+1, (k*x)^k/(1-2*k*x)^k*exp(-k*x/(1-2*k*x+x*O(x^n)))/k!), n)}
for(n=0, 25, print1(a(n), ", "))

Formula

O.g.f.: Sum_{n>=0} (n*x)^n/(1-2*n*x)^n * exp(-n*x/(1-2*n*x)) / n!.

A245060 a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(n,k) * 3^(n-k) for n>0, a(0)=1, where S2(n,k) = A048993(n,k) are Stirling numbers of the 2nd kind.

Original entry on oeis.org

1, 1, 4, 28, 271, 3172, 43174, 666577, 11445214, 215478712, 4401799930, 96757165012, 2273105615356, 56755763435503, 1499039156935948, 41714498328290992, 1218787798107634291, 37275555462806318512, 1190200470204107432854, 39581409916012393962280, 1368112674516484881342244
Offset: 0

Views

Author

Paul D. Hanna, Jul 10 2014

Keywords

Examples

			O.g.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 271*x^4 + 3172*x^5 + 43174*x^6 +...
where
A(x) = 1 + x/(1-3*x)*exp(-x/(1-3*x)) + 2^2*x^2/(1-6*x)^2*exp(-2*x/(1-6*x))/2! + 3^3*x^3/(1-9*x)^3*exp(-3*x/(1-9*x))/3! + 4^4*x^4/(1-12*x)^4*exp(-4*x/(1-12*x))/4! +...
simplifies to a power series in x with integer coefficients.
Illustrate the definition of the terms by:
a(2) = 1*1*3 + 1*1 = 4;
a(3) = 1*1*3^2 + 2*3*3 + 1*1 = 28;
a(4) = 1*1*3^3 + 3*7*3^2 + 3*6*3 + 1*1 = 271;
a(5) = 1*1*3^4 + 4*15*3^3 + 6*25*3^2 + 4*10*3 + 1*1 = 3172;
a(6) = 1*1*3^5 + 5*31*3^4 + 10*90*3^3 + 10*65*3^2 + 5*15*3 + 1*1 = 43174; ...

Crossrefs

Cf. A134055, A245059, A218667, A218670.

Programs

  • PARI
    {a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1)*polcoeff(1/prod(i=0, k, 1-i*x +x*O(x^(n-k))), n-k)*3^(n-k)))}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff(sum(k=0, n+1, (k*x)^k/(1-3*k*x)^k*exp(-k*x/(1-3*k*x+x*O(x^n)))/k!), n)}
for(n=0, 25, print1(a(n), ", "))

Formula

O.g.f.: Sum_{n>=0} (n*x)^n/(1-3*n*x)^n * exp(-n*x/(1-3*n*x)) / n!.

A242019 a(n) = Sum_{k=0..n} Stirling2(2*n+k, k) * C(n, k).

Original entry on oeis.org

1, 1, 33, 3409, 728575, 265362370, 147228369351, 115651594418010, 122167455441632423, 167035663137431205196, 287018982366654934570328, 605456750773492887086145669, 1538306721887736189212800143193, 4633572348321634923252339927247392
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 11 2014

Keywords

Crossrefs

Cf. A243942, A218667.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k] * StirlingS2[2*n+k,k],{k,0,n}],{n,0,20}]

Formula

a(n) ~ c * (r^4/((1-r)*(2*r-1)^2))^n * n^(2*n-1/2) / exp(2*n), where r = 0.949867370961706500554205072094811326960829788646... is the root of the equation (1-r)*(2+r)/r^2 = -LambertW(-exp(-1-2/r)*(2+r)/r), and c = 0.42307980713011095154197903821771057626302758607...
Previous Showing 11-15 of 15 results.

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