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 1-9 of 9 results.

A244116 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 1 as Sum_{k=0..n} T(n,k)*binomial(n,k).

Original entry on oeis.org

1, 0, 1, 0, 1, -1, 0, 1, -2, 4, 0, 1, -4, 12, -27, 0, 1, -8, 36, -108, 256, 0, 1, -16, 108, -432, 1280, -3125, 0, 1, -32, 324, -1728, 6400, -18750, 46656, 0, 1, -64, 972, -6912, 32000, -112500, 326592, -823543, 0, 1, -128, 2916, -27648, 160000, -675000, 2286144, -6588344, 16777216
Offset: 0

Views

Author

Stanislav Sykora, Jun 21 2014

Keywords

Comments

T(n,k) = (1-k)^(k-1) * k^(n-k) for k>0, and T(n,0) = 0^n by convention.

Examples

			The first few rows of the triangle are:
  1
  0 1
  0 1 -1
  0 1 -2 4
  0 1 -4 12  -27
  0 1 -8 36 -108 256
  ...

Links

Crossrefs

Cf. A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.
Cf. A357247.

Programs

  • Maple
    A244116 := (n, j) -> (-1)^(j + 1) * j^(n - j) * (j - 1)^(j - 1):
for n from 0 to 9 do seq(A244116(n, k), k = 0..n) od; # Peter Luschny, Jan 28 2023
  • PARI
    seq(nmax,b)={my(v,n,k,irow);
  v = vector((nmax+1)*(nmax+2)/2);v[1]=1;
  for(n=1,nmax,irow=1+n*(n+1)/2;v[irow]=0;
    for(k=1,n,v[irow+k] = (1-k*b)^(k-1)*(k*b)^(n-k););
  );return(v);}
  a=seq(100,1);

A273954 E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * exp(n*x) * A(x)^n.

Original entry on oeis.org

1, 1, 5, 37, 393, 5481, 95053, 1975821, 47939601, 1330923601, 41629292181, 1448989481589, 55561575788953, 2327512861252281, 105767732851318749, 5182512561142513501, 272391086209524010017, 15287595381259195453089, 912525533175190887597349, 57726267762799335649572549
Offset: 0

Views

Author

Paul D. Hanna, Jun 14 2016

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 393*x^4/4! + 5481*x^5/5! + 95053*x^6/6! + 1975821*x^7/7! + 47939601*x^8/8! + 1330923601*x^9/9! + 41629292181*x^10/10! + 1448989481589*x^11/11! + 55561575788953*x^12/12! +...
such that
A(x) = 1 + x*exp(x)*A(x) + x^2/2!*exp(2*x)*A(x)^2 + x^3/3!*exp(3*x)*A(x)^3 + x^4/4!*exp(4*x)*A(x)^4 + x^5/5!*exp(5*x)*A(x)^5 + x^6/6!*exp(6*x)*A(x)^6 +...
The logarithm of A(x) begins:
log(A(x)) = x + 4*x^2/2! + 24*x^3/3! + 224*x^4/4! + 2880*x^5/5! + 47232*x^6/6! + 942592*x^7/7! + 22171648*x^8/8! + 600698880*x^9/9! + 18422374400*x^10/10! +...+ A216857(n)*x^n/n! +...
which equals -LambertW(-x*exp(x)).

Links

Crossrefs

Cf. A273953, A216857, A357247, A360176 (column 1 unsigned).
Cf. A161628, A244119.

Programs

  • Maple
    A273954 := n -> add(binomial(n, j) * j^(n - j) * (j + 1)^(j - 1), j = 0..n):
seq(A273954(n), n = 0..24); # Peter Luschny, Jan 29 2023
  • Mathematica
    CoefficientList[Series[-LambertW[-x*E^x] / (x*E^x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jun 23 2016 *)
  • PARI
    {a(n) = my(A=1+x); for(i=1,n, A = sum(m=0,n,x^m/m!*exp(m*x +x*O(x^n))*A^m) ); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    x='x+O('x^50); Vec(serlaplace(-lambertw(-x*exp(x))/(x*exp(x)))) \\ G. C. Greubel, Nov 16 2017
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x*exp(x))^k/k!))) \\ Seiichi Manyama, Feb 08 2023

Formula

E.g.f.: -LambertW(-x*exp(x)) / (x*exp(x)). [corrected by Vaclav Kotesovec, Jun 23 2016]
E.g.f.: exp( L(x) ) where L(x) = -LambertW(-x*exp(x)) is the e.g.f. of A216857.
a(n) ~ sqrt(1+LambertW(exp(-1))) * n^(n-1) / (exp(n-1) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Jun 23 2016
E.g.f.: A(x) = exp(x*exp(x)*A(x)). - Alexander Burstein, Aug 11 2018
From Peter Luschny, Jan 29 2023: (Start)
a(n) = Sum_{j=0..n} binomial(n, j) * j^(n - j) * (j + 1)^(j - 1).
a(n) = Sum_{k=0..n} (-1)^k*A161628(n, k).
a(n) = Sum_{k=0..n} (-1)^(n-k)*A244119(n, k). (End)

A380427 E.g.f. A(x) satisfies A(x) = exp( -x/A(x) * exp(x/A(x)) ).

Original entry on oeis.org

1, -1, -3, -19, -211, -3301, -66581, -1643587, -47986247, -1617313033, -61796668969, -2639583958111, -124635062782187, -6446216079166189, -362427406400015165, -22008570202561166491, -1435560535563493528591, -100100185675457848764433, -7430481272601559979203409
Offset: 0

Views

Author

Seiichi Manyama, Jan 24 2025

Keywords

Links

Crossrefs

Cf. A162695, A380425, A380426.
Cf. A357247.

Formula

E.g.f.: 1/B(x), where B(x) is the e.g.f. of A162695.
a(n) = -Sum_{k=1..n} k^(n-k) * (n-1)^(k-1) * binomial(n,k) for n > 0.

A360465 E.g.f. satisfies A(x) = exp(x * exp(2*x) * A(x)).

Original entry on oeis.org

1, 1, 7, 64, 829, 14056, 295399, 7426252, 217637305, 7291538704, 275050426411, 11540336658676, 533224609095061, 26908386824872216, 1472691380336896399, 86892807951798473116, 5498668489586321670769, 371511527654280649783840
Offset: 0

Views

Author

Seiichi Manyama, Feb 08 2023

Keywords

Links

Crossrefs

Cf. A038050, A273954, A357247, A360466.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*exp(2*x)))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(-lambertw(-x*exp(2*x))/(x*exp(2*x))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x*exp(2*x))^k/k!)))
  • PARI
    a(n) = sum(k=0, n, (2*k)^(n-k)*(k+1)^(k-1)*binomial(n, k));

Formula

E.g.f.: A(x) = exp( -LambertW(-x * exp(2*x)) ).
E.g.f.: A(x) = -LambertW(-x * exp(2*x)) / (x * exp(2*x)).
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (x * exp(2*x))^k / k!.
a(n) = Sum_{k=0..n} (2*k)^(n-k) * (k+1)^(k-1) * binomial(n,k).
a(n) ~ sqrt(1+LambertW(2*exp(-1))) * 2^n * n^(n-1) / (exp(n-1) * LambertW(2*exp(-1))^n). - Vaclav Kotesovec, Feb 08 2023

A380406 E.g.f. satisfies A(x) = exp( 2 * x * exp(x) * A(x)^(1/2) ).

Original entry on oeis.org

1, 2, 12, 104, 1232, 18592, 342208, 7451264, 187631872, 5369721344, 172255038464, 6125052946432, 239195824279552, 10179739052908544, 469024768235192320, 23263095316577681408, 1235978286454556131328, 70040404736026578386944, 4217180561907991530176512
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2025

Keywords

Links

Crossrefs

Cf. A273954, A357247, A380407.
Cf. A273953, A360473.

Programs

  • PARI
    a(n) = 2*sum(k=0, n, k^(n-k)*(k+2)^(k-1)*binomial(n, k));

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A273954.
E.g.f.: A(x) = exp( -2*LambertW(-x * exp(x)) ).
a(n) = 2 * Sum_{k=0..n} k^(n-k) * (k+2)^(k-1) * binomial(n,k).
a(n) ~ 2 * sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (exp(n-2) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Aug 05 2025

A380407 E.g.f. satisfies A(x) = exp( 3 * x * exp(x) * A(x)^(1/3) ).

Original entry on oeis.org

1, 3, 21, 207, 2697, 43803, 854685, 19512615, 510977937, 15112457523, 498560461989, 18160560320895, 724240913035545, 31394996915447883, 1470245245400432685, 73987438021589516247, 3982389565847576723745, 228331703268783136636515, 13894569264190369648271157
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2025

Keywords

Links

Crossrefs

Cf. A273954, A357247, A380406.

Programs

  • Mathematica
    terms = 19; A[] = 0; Do[A[x] = Exp[3*x*Exp[x]*A[x]^(1/3)] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Jun 14 2025 *)
  • PARI
    a(n) = 3*sum(k=0, n, k^(n-k)*(k+3)^(k-1)*binomial(n, k));

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A273954.
E.g.f.: A(x) = exp( -3*LambertW(-x * exp(x)) ).
a(n) = 3 * Sum_{k=0..n} k^(n-k) * (k+3)^(k-1) * binomial(n,k).

A357246 E.g.f. satisfies A(x) * log(A(x)) = (1-x) * (exp(x) - 1).

Original entry on oeis.org

1, 1, -2, 5, -49, 497, -6926, 116510, -2325422, 53538315, -1397740279, 40792008435, -1316056239994, 46509292766172, -1786748828967402, 74139054468535061, -3304409577659864305, 157444695280699565069, -7986085592316390890618, 429645521271113815480246
Offset: 0

Views

Author

Seiichi Manyama, Sep 19 2022

Keywords

Links

Crossrefs

Cf. A356902, A357243, A357247.

Programs

  • Mathematica
    nmax = 19; A[_] = 1;
Do[A[x_] = Exp[-(((Exp[x]-1)*(x-1))/A[x])]+O[x]^(nmax+1)//Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*((1-x)*(exp(x)-1))^k/k!)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw((1-x)*(exp(x)-1)))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((1-x)*(exp(x)-1)/lambertw((1-x)*(exp(x)-1))))

Formula

E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * ((1-x) * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( LambertW((1-x) * (exp(x) - 1)) ).
E.g.f.: A(x) = (1-x) * (exp(x) - 1)/LambertW((1-x) * (exp(x) - 1)).

A359759 Table read by rows. T(n, k) = (-1)^(n - k) * Sum_{j=k..n} binomial(n, j) * A354794(j, k) * j^(n - j).

Original entry on oeis.org

1, 0, 1, 0, -3, 1, 0, 13, -9, 1, 0, -103, 79, -18, 1, 0, 1241, -905, 265, -30, 1, 0, -19691, 13771, -4290, 665, -45, 1, 0, 384805, -262885, 82621, -14630, 1400, -63, 1, 0, -8918351, 6007247, -1888362, 353381, -40390, 2618, -84, 1
Offset: 0

Views

Author

Peter Luschny, Jan 27 2023

Keywords

Comments

Inspired by a formula of Mélika Tebni in A048993.

Examples

			Triangle T(n, k) starts:
[0] 1;
[1] 0,         1;
[2] 0,        -3,          1;
[3] 0,        13,         -9,        1;
[4] 0,      -103,         79,      -18,        1;
[5] 0,      1241,       -905,      265,      -30,       1;
[6] 0,    -19691,      13771,    -4290,      665,     -45,      1;
[7] 0,    384805,    -262885,    82621,   -14630,    1400,    -63,    1;
[8] 0,  -8918351,    6007247, -1888362,   353381,  -40390,   2618,  -84,    1;
[9] 0, 238966705, -159432369, 50110705, -9627702, 1206471, -96138, 4494, -108, 1;

Crossrefs

Cf. A059297, A354794, A357247, A048993.

Programs

  • Maple
    T := (n, k) -> (-1)^(n - k)*add(binomial(n, j) * A354794(j, k) * j^(n - j), j = k..n): for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

Formula

E.g.f. of column k: (exp(LambertW(x*exp(-x))) - 1)^k / k!. (Note that (exp(-LambertW(-x*exp(-x))) - 1)^k / k! is the e.g.f. of column k of Stirling2.) - Mélika Tebni, Jan 27 2023

A360466 E.g.f. satisfies A(x) = exp(2 * x * exp(x) * A(x)).

Original entry on oeis.org

1, 2, 16, 206, 3832, 93962, 2871820, 105355406, 4515648784, 221598121490, 12257187851284, 754703476252310, 51204818674338328, 3796079000648275226, 305328667748448560668, 26483633169003911205278, 2464307301750079915255840, 244872778601760932275686434
Offset: 0

Views

Author

Seiichi Manyama, Feb 08 2023

Keywords

Links

Crossrefs

Cf. A273954, A357247, A360465.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x*exp(x)))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(-lambertw(-2*x*exp(x))/(2*x*exp(x))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(2*x*exp(x))^k/k!)))
  • PARI
    a(n) = sum(k=0, n, 2^k*k^(n-k)*(k+1)^(k-1)*binomial(n, k));

Formula

E.g.f.: A(x) = exp( -LambertW(-2 * x * exp(x)) ).
E.g.f.: A(x) = -LambertW(-2 * x * exp(x)) / (2 * x * exp(x)).
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (2 * x * exp(x))^k / k!.
a(n) = Sum_{k=0..n} 2^k * k^(n-k) * (k+1)^(k-1) * binomial(n,k).
a(n) ~ sqrt(1+LambertW(exp(-1)/2)) * n^(n-1) / (exp(n-1) * LambertW(exp(-1)/2)^n). - Vaclav Kotesovec, Feb 08 2023
Showing 1-9 of 9 results.

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