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 1-10 of 16 results. Next

A218672 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n*x)^n/n! * exp(-n*x*A(n*x)).

Original entry on oeis.org

1, 1, 2, 9, 63, 659, 9833, 206961, 6133990, 256650268, 15213478000, 1281205909177, 153588353066135, 26245044813624300, 6399076697684238375, 2227912079081482302977, 1108302173165578509079527, 788171767077184315422131588, 801638519723021288783092512047
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

Compare to the LambertW identities:
(1) Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).
(2) Sum_{n>=0} n^n * x^n * C(x)^n/n! * exp(-n*x*C(x)) = C(x), where C(x) = 1 + x*C(x)^2 is the o.g.f. of the Catalan numbers (A000108).

Examples

			O.g.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 63*x^4 + 659*x^5 + 9833*x^6 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(2*x)^2/2!*exp(-2*x*A(2*x)) + 3^3*x^3*A(3*x)^3/3!*exp(-3*x*A(3*x)) + 4^4*x^4*A(4*x)^4/4!*exp(-4*x*A(4*x)) + 5^5*x^5*A(5*x)^5/5!*exp(-5*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.

Links

Crossrefs

Cf. A218673, A218674, A218675, A218676, A218681.
Cf. A217900, A218670, A218667, A218668, A218669, A134055.
Cf. A193363, A221409, A221410, A221411, A221412, A221413.
Cf. A209276, A209277.

Programs

  • Mathematica
    a[n_] := Module[{A}, A[x_] = 1 + x; For[i = 1, i <= n, i++, A[x_] = Sum[If[k == 0, 1, k^k] x^k A[k x]^k/k! Exp[-k x A[k x] + x O[x]^i] // Normal, {k, 0, n}]]; Coefficient[ A[x], x, n]];
a /@ Range[0, 18] (* Jean-François Alcover, Sep 29 2019 *)
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A,x,k*x)^k/k!*exp(-k*x*subst(A,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A218667 O.g.f.: Sum_{n>=0} 1/(1-n*x)^n * x^n/n! * exp(-x/(1-n*x)).

Original entry on oeis.org

1, 0, 1, 1, 4, 13, 46, 181, 778, 3585, 17566, 91171, 499324, 2873839, 17317743, 108933098, 713481122, 4855161425, 34257461754, 250177938679, 1887886966690, 14699340919293, 117933068390123, 973776266303732, 8265721830953558, 72052688932613079, 644393453082317301
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

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

Examples

			O.g.f.: A(x) = 1 + x^2 + x^3 + 4*x^4 + 13*x^5 + 46*x^6 + 181*x^7 +...
where
A(x) = exp(-x) + x/(1-x)*exp(-x/(1-x)) + x^2/(1-2*x)^2/2!*exp(-x/(1-2*x)) + x^3/(1-3*x)^3/3!*exp(-x/(1-3*x)) + x^4/(1-4*x)^4/4!*exp(-x/(1-4*x)) + x^5/(1-5*x)^5/5!*exp(-x/(1-5*x)) + x^6/(1-6*x)^6/6!*exp(-x/(1-6*x)) +...
simplifies to a power series in x with integer coefficients.

Links

Crossrefs

Cf. A245111, A245110, A218668, A218669, A218670, A185040, A217900.

Programs

  • PARI
    {a(n)=local(A=1+x,X=x+x*O(x^n));A=sum(k=0,n,1/(1-k*X)^k*x^k/k!*exp(-X/(1-k*X)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* From a(n) = Sum_{k=1..n} Stirling2(n-k, k) * C(-1, k-1) */
{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} Stirling2(n-k, k) * C(n-1, k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jul 30 2014
Antidiagonal sums of Triangle A245111.

A134055 a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(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, 2, 8, 41, 252, 1782, 14121, 123244, 1169832, 11960978, 130742196, 1518514076, 18645970943, 241030821566, 3268214127548, 46338504902485, 685145875623056, 10538790233183702, 168282662416550040, 2784205185437851772, 47646587512911994120
Offset: 0

Views

Author

Paul D. Hanna, Oct 08 2007

Keywords

Examples

			O.g.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 41*x^4 + 252*x^5 + 1782*x^6 + 14121*x^7 +...
where
A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^2*x^2/(1-2*x)^2*exp(-2*x/(1-2*x))/2! + 3^3*x^3/(1-3*x)^3*exp(-3*x/(1-3*x))/3! + 4^4*x^4/(1-4*x)^4*exp(-4*x/(1-4*x))/4! +...
simplifies to a power series in x with integer coefficients.
Illustrate the definition of the terms by:
a(4) = 1*1 + 3*7 + 3*6 + 1*1 = 41;
a(5) = 1*1 + 4*15 + 6*25 + 4*10 + 1*1 = 252;
a(6) = 1*1 + 5*31 + 10*90 + 10*65 + 5*15 + 1*1 = 1782.

Links

Crossrefs

Cf. A037011, A048297, A048993, A122455, A218667, A218670, A245059, A245060.

Programs

  • Maple
    a:= proc(n) option remember; local b; b:=
      proc(h, m) option remember; `if`(h=0,
        binomial(n-1, m-1), m*b(h-1, m)+b(h-1, m+1) )
      end; b(n, 0)
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Jun 24 2023
  • Mathematica
    Flatten[{1,Table[Sum[Binomial[n-1,k-1] * StirlingS2[n,k],{k,1,n}],{n,1,20}]}] (* Vaclav Kotesovec, Aug 11 2014 *)
  • 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)))
  • PARI
    a(n)=polcoeff(sum(k=0,n+1,(k*x)^k/(1-k*x)^k*exp(-k*x/(1-k*x+x*O(x^n)))/k!),n)
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Nov 04 2012

Formula

O.g.f.: Sum_{n>=0} (n*x)^n/(1-n*x)^n * exp(-n*x/(1-n*x)) / n!. - Paul D. Hanna, Nov 04 2012
From Alois P. Heinz, Jun 24 2023: (Start)
a(n) mod 2 = A037011(n) for n >= 1.
a(n) mod 2 = 1 <=> n in { A048297 } or n = 0. (End)

Extensions

An initial '1' was added and definition changed slightly by Paul D. Hanna, Nov 04 2012

A218668 O.g.f.: Sum_{n>=0} 1/(1-n^2*x)^n * x^n/n! * exp(-x/(1-n^2*x)).

Original entry on oeis.org

1, 0, 1, 3, 22, 161, 1546, 18857, 270320, 4471693, 85455574, 1865128265, 45735737037, 1247518965519, 37654095184226, 1250673144714138, 45415758777730668, 1792734161930717221, 76595370803745016626, 3529261203030717032927, 174742139545017029583279
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

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

Examples

			O.g.f.: A(x) = 1 + x^2 + 3*x^3 + 22*x^4 + 161*x^5 + 1546*x^6 + 18857*x^7 +...
where
A(x) = exp(-x) + x/(1-x)*exp(-x/(1-x)) + x^2/(1-4*x)^2/2!*exp(-x/(1-4*x)) + x^3/(1-9*x)^3/3!*exp(-x/(1-9*x)) + x^4/(1-16*x)^4/4!*exp(-x/(1-16*x)) + x^5/(1-25*x)^5/5!*exp(-x/(1-25*x)) +...
simplifies to a power series in x with integer coefficients.

Links

Crossrefs

Cf. A218667, A218669, A218670, A217900.

Programs

  • PARI
    {a(n)=local(A=1+x,X=x+x*O(x^n));A=sum(k=0,n,1/(1-k^2*X)^k*x^k/k!*exp(-X/(1-k^2*X)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A218669 O.g.f.: Sum_{n>=0} 1/(1-n^3*x)^n * x^n/n! * exp(-x/(1-n^3*x)).

Original entry on oeis.org

1, 0, 1, 7, 97, 1561, 41136, 1551814, 72440460, 4281320257, 324623105584, 30086950057627, 3299720918091511, 428431079916572044, 65637957066642609845, 11659659637028895337265, 2367270866164121777222596, 546795407830461739380895161, 143176487805296033192642234802
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

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

Examples

			O.g.f.: A(x) = 1 + x^2 + 7*x^3 + 97*x^4 + 1561*x^5 + 41136*x^6 +...
where
A(x) = exp(-x) + x/(1-x)*exp(-x/(1-x)) + x^2/(1-8*x)^2/2!*exp(-x/(1-8*x)) + x^3/(1-27*x)^3/3!*exp(-x/(1-27*x)) + x^4/(1-64*x)^4/4!*exp(-x/(1-64*x)) + x^5/(1-125*x)^5/5!*exp(-x/(1-125*x)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218667, A218668, A218670, A217900.

Programs

  • PARI
    {a(n)=local(A=1+x,X=x+x*O(x^n));A=sum(k=0,n,1/(1-k^3*X)^k*x^k/k!*exp(-X/(1-k^3*X)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A218673 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(2*n)/n! * exp(-n*x*A(n*x)^2).

Original entry on oeis.org

1, 1, 3, 20, 209, 3173, 67292, 1970761, 79764057, 4490097388, 354111363537, 39360693851404, 6193012446752244, 1383433132321835172, 439684769985895688173, 199116777197880585373014, 128631139424158036273736167, 118640007280899188486618513612
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

Compare to the LambertW identity:
Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 3*x^2 + 20*x^3 + 209*x^4 + 3173*x^5 + 67292*x^6 +...
where
A(x) = 1 + x*A(x)^2*exp(-x*A(x)^2) + 2^2*x^2*A(2*x)^4/2!*exp(-2*x*A(2*x)^2) + 3^3*x^3*A(3*x)^6/3!*exp(-3*x*A(3*x)^2) + 4^4*x^4*A(4*x)^8/4!*exp(-4*x*A(4*x)^2) + 5^5*x^5*A(5*x)^10/5!*exp(-5*x*A(5*x)^2) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A218674, A218675, A218676.
Cf. A217900, A218670, A218667, A218668, A218669, A134055.

Programs

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

A218674 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(3*n)/n! * exp(-n*x*A(n*x)^3).

Original entry on oeis.org

1, 1, 4, 34, 455, 8710, 230077, 8285224, 407456797, 27587687551, 2596034329278, 342275007167359, 63606742005546232, 16730509857101195808, 6246818082857455197662, 3317816101992338134691233, 2510420393373091580780786808, 2709148467943025007607468405672
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

Compare to the LambertW identity:
Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 4*x^2 + 34*x^3 + 455*x^4 + 8710*x^5 + 230077*x^6 +...
where
A(x) = 1 + x*A(x)^3*exp(-x*A(x)^3) + 2^2*x^2*A(2*x)^6/2!*exp(-2*x*A(2*x)^3) + 3^3*x^3*A(3*x)^9/3!*exp(-3*x*A(3*x)^3) + 4^4*x^4*A(4*x)^12/4!*exp(-4*x*A(4*x)^3) + 5^5*x^5*A(5*x)^15/5!*exp(-5*x*A(5*x)^3) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A218673, A218675, A218676.
Cf. A217900, A218670, A218667, A218668, A218669, A134055.

Programs

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

A218675 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(4*n)/n! * exp(-n*x*A(n*x)^4).

Original entry on oeis.org

1, 1, 5, 51, 817, 18562, 576687, 24203258, 1375038677, 106708683355, 11435867474152, 1708844338589752, 358640659116617571, 106261016900832212139, 44607231638918264608274, 26598477338494285370797703, 22569718290467849884279856477
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

Compare to the LambertW identity:
Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 5*x^2 + 51*x^3 + 817*x^4 + 18562*x^5 + 576687*x^6 +...
where
A(x) = 1 + x*A(x)^4*exp(-x*A(x)^4) + 2^2*x^2*A(2*x)^8/2!*exp(-2*x*A(2*x)^4) + 3^3*x^3*A(3*x)^12/3!*exp(-3*x*A(3*x)^4) + 4^4*x^4*A(4*x)^16/4!*exp(-4*x*A(4*x)^4) + 5^5*x^5*A(5*x)^20/5!*exp(-5*x*A(5*x)^4) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A218673, A218674, A218676.
Cf. A217900, A218670, A218667, A218668, A218669, A134055.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A^4,x,k*x)^k/k!*exp(-k*x*subst(A^4,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A218676 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(5*n)/n! * exp(-n*x*A(n*x)^5).

Original entry on oeis.org

1, 1, 6, 71, 1311, 34146, 1207717, 57298282, 3653975784, 316252925221, 37596625187796, 6206102367103899, 1434418185304457039, 466995106832397752352, 215051811411620578152401, 140491107719613466192347681, 130481943378389095603359529403
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

Compare to the LambertW identity:
Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 6*x^2 + 71*x^3 + 1311*x^4 + 34146*x^5 + 1207717*x^6 +...
where
A(x) = 1 + x*A(x)^5*exp(-x*A(x)^5) + 2^2*x^2*A(2*x)^10/2!*exp(-2*x*A(2*x)^5) + 3^3*x^3*A(3*x)^15/3!*exp(-3*x*A(3*x)^5) + 4^4*x^4*A(4*x)^20/4!*exp(-4*x*A(4*x)^5) + 5^5*x^5*A(5*x)^25/5!*exp(-5*x*A(5*x)^5) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A218673, A218674, A218675.
Cf. A217900, A218670, A218667, A218668, A218669, A134055.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A^5,x,k*x)^k/k!*exp(-k*x*subst(A^5,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A218677 O.g.f.: Sum_{n>=0} n^n * (1+n*x)^(2*n) * x^n/n! * exp(-n*x*(1+n*x)^2).

Original entry on oeis.org

1, 1, 3, 14, 79, 516, 3802, 30668, 268815, 2522594, 25201736, 266014607, 2953171684, 34326755191, 416313253084, 5251970372080, 68737673434847, 931207966502919, 13031639620371226, 188051624603419973, 2793741995189126920, 42668132798523737471, 669061042470049870917
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

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

Examples

			O.g.f.: A(x) = 1 + x + 3*x^2 + 14*x^3 + 79*x^4 + 516*x^5 + 3802*x^6 +...
where
A(x) = 1 + (1+x)^2*x*exp(-x*(1+x)^2) + 2^2*(1+2*x)^4*x^2/2!*exp(-2*x*(1+2*x)^2) + 3^3*(1+3*x)^6*x^3/3!*exp(-3*x*(1+3*x)^2) + 4^4*(1+4*x)^8*x^4/4!*exp(-4*x*(1+4*x)^2) + 5^5*(1+5*x)^10*x^5/5!*exp(-5*x*(1+5*x)^2) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218670, A218678, A218679.

Programs

  • PARI
    {a(n)=local(A=1+x);A=sum(k=0,n,k^k*(1+k*x)^(2*k)*x^k/k!*exp(-k*x*(1+k*x)^2+x*O(x^n)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
Showing 1-10 of 16 results. Next

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