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 19 results. Next

A244119 Triangle read by rows: terms of a binomial decomposition of 1 as Sum(k=0..n)T(n,k).

Original entry on oeis.org

1, 0, 1, 0, -2, 3, 0, 3, -18, 16, 0, -4, 72, -192, 125, 0, 5, -240, 1440, -2500, 1296, 0, -6, 720, -8640, 30000, -38880, 16807, 0, 7, -2016, 45360, -280000, 680400, -705894, 262144, 0, -8, 5376, -217728, 2240000, -9072000, 16941456, -14680064, 4782969
Offset: 0

Views

Author

Stanislav Sykora, Jun 21 2014

Keywords

Comments

T(n,k)=(1+k)^(k-1)*(-k)^(n-k)*binomial(n,k) for k>0, while T(n,0)=0^n by convention.
Sequence A161628, arising from a different context, appears to be the same, but with opposite signs of odd rows.

Examples

			First rows of the triangle, all summing up to 1:
1
0  1
0 -2    3
0  3  -18   16
0 -4   72 -192   125
0  5 -240 1440 -2500 1296

Links

Crossrefs

Cf. A161628, A244116, A244117, A244118, 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. A273954.

Programs

  • Maple
    A244119 := (n, k) -> (1+k)^(k-1)*(-k)^(n-k)*binomial(n,k):
seq(seq(A244119(n, k), k = 0..n), n = 0..8); # Peter Luschny, Jan 29 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)*binomial(n,k););
  );return(v);}
  a=seq(100,-1);

A360473 E.g.f. satisfies A(x) = exp( x * exp(x) * A(x)^2 ).

Original entry on oeis.org

1, 1, 7, 82, 1441, 34036, 1013149, 36446698, 1538703457, 74607811048, 4086635087701, 249593193648646, 16819085803158577, 1239637405609740268, 99206330021667838285, 8567230421555333516746, 794104205843228382969409
Offset: 0

Views

Author

Seiichi Manyama, Feb 08 2023

Keywords

Links

Crossrefs

Cf. A273954, A360465, A360474.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x*Exp[x]*A[x]^2] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = sum(k=0, n, k^(n-k)*(2*k+1)^(k-1)*binomial(n, k));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x*exp(x))/2)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(-lambertw(-2*x*exp(x))/(2*x*exp(x)))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(sum(k=0, N, (k+1)^(k-1)*(2*x*exp(x))^k/k!))))

Formula

a(n) = Sum_{k=0..n} k^(n-k) * (2*k+1)^(k-1) * binomial(n,k).
E.g.f.: A(x) = exp( -LambertW(-2*x * exp(x))/2 ).
E.g.f.: A(x) = sqrt( -LambertW(-2*x * exp(x)) / (2*x * exp(x)) ).
E.g.f.: A(x) = sqrt( Sum_{k>=0} (k+1)^(k-1) * (2*x * exp(x))^k / k! ).
a(n) ~ sqrt(1 + LambertW(exp(-1)/2)) * n^(n-1) / (2 * exp(n - 1/2) * LambertW(exp(-1)/2)^n). - Vaclav Kotesovec, Feb 17 2023

A273953 E.g.f. satisfies A(x) = Sum_{n>=0} x^n/n! * exp(n/2*x) * A(x)^(n/2).

Original entry on oeis.org

1, 1, 3, 13, 77, 581, 5347, 58213, 732937, 10487737, 168217811, 2990748509, 58397418037, 1242643927357, 28627000014355, 709933328752981, 18859531958840273, 534365880859577777, 16087267158157316323, 512844446937529664173, 17259468942471032848861, 611530055485070740134901, 22755171133646348369448323, 887228501593124485460914373, 36173480392953890421156056665, 1539307965110263598673884269801, 68247672532254821767545000249907
Offset: 0

Views

Author

Paul D. Hanna, Jun 14 2016

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 77*x^4/4! + 581*x^5/5! + 5347*x^6/6! + 58213*x^7/7! + 732937*x^8/8! + 10487737*x^9/9! + 168217811*x^10/10! + 2990748509*x^11/11! + 58397418037*x^12/12! +...
such that
A(x) = 1 + x*exp(x/2)*A(x)^(1/2) + x^2/2!*exp(x)*A(x) + x^3/3!*exp(3*x/2)*A(x)^(3/2) + x^4/4!*exp(2*x)*A(x)^2 + x^5/5!*exp(5*x/2)*A(x)^(5/2) + x^6/6!*exp(3*x)*A(x)^3 +...
The logarithm of A(x) begins:
log(A(x)) = x + 2*x^2/2! + 6*x^3/3! + 28*x^4/4! + 180*x^5/5! + 1476*x^6/6! + 14728*x^7/7! + 173216*x^8/8! + 2346480*x^9/9! + 35981200*x^10/10! + 616111056*x^11/11! + 11652662880*x^12/12! +...+ A100526(n)*x^n/n! +...
which equals -2*LambertW(-x*exp(x/2)/2).

Links

Crossrefs

Cf. A273954, A100526.

Programs

  • Mathematica
    CoefficientList[Series[4*LambertW[-x/2*E^(x/2)]^2 / (x^2*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/2*x +x*O(x^n))*A^(m/2)) ); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    a(n) = sum(k=0, n, k^(n-k)*(k+2)^(k-1)*binomial(n, k))/2^(n-1); \\ Seiichi Manyama, Feb 11 2023

Formula

E.g.f.: 4*LambertW(-x/2*exp(x/2))^2 / (x^2*exp(x)).
E.g.f.: exp( L(x) ) where L(x) = -2*LambertW(-x*exp(x/2)/2) is the e.g.f. of A100526.
a(n) ~ sqrt(1+LambertW(exp(-1)))*n^(n-1)/(2^(n-1)*exp(n-2)*LambertW(exp(-1))^n). - Vaclav Kotesovec, Jun 23 2016
From Seiichi Manyama, Feb 11 2023: (Start)
E.g.f. satisfies A(x) = exp( x * ( exp(x) * A(x) )^(1/2) ).
a(n) = (1/2^(n-1)) * Sum_{k=0..n} k^(n-k) * (k+2)^(k-1) * binomial(n,k). (End)

A360547 E.g.f. satisfies A(x) = exp( x * ( exp(x) * A(x) )^2 ).

Original entry on oeis.org

1, 1, 9, 121, 2417, 64721, 2180665, 88719625, 4233968737, 231991022881, 14356691152361, 990506937621785, 75390334060230865, 6275675303410022641, 567191776288882702105, 55313848534122299876521, 5789703106014903009828545, 647414950001156861671249985
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2023

Keywords

Links

Crossrefs

Cf. A273953, A273954, A360544.
Cf. A360465, A360473.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x*(Exp[x]*A[x])^2] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x*exp(2*x))/2)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(-lambertw(-2*x*exp(2*x))/(2*x*exp(2*x)))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(sum(k=0, N, (k+1)^(k-1)*(2*x*exp(2*x))^k/k!))))
  • PARI
    a(n) = sum(k=0, n, (2*k)^(n-k)*(2*k+1)^(k-1)*binomial(n, k));

Formula

E.g.f.: A(x) = exp( (-1/2) * LambertW(-2*x * exp(2*x)) ).
E.g.f.: A(x) = sqrt( -LambertW(-2*x * exp(2*x)) / (2*x * exp(2*x)) ).
E.g.f.: A(x) = sqrt( Sum_{k>=0} (k+1)^(k-1) * (2*x * exp(2*x))^k / k! ).
a(n) = Sum_{k=0..n} (2*k)^(n-k) * (2*k+1)^(k-1) * binomial(n,k).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 2^(n-1) * n^(n-1) / (exp(n - 1/2) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Feb 17 2023

A360544 E.g.f. satisfies A(x) = exp( x * ( exp(x) * A(x) )^(3/2) ).

Original entry on oeis.org

1, 1, 7, 73, 1117, 22741, 580159, 17826985, 641494249, 26473635865, 1232945359111, 63978649829161, 3660871368065509, 229016870623703917, 15550838554432967647, 1139139301403727884521, 89544381521098908259729, 7518611017848248249471089
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2023

Keywords

Links

Crossrefs

Cf. A273953, A273954, A360547.
Cf. A360473, A360545.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2*lambertw(-3/2*x*exp(3*x/2))/3)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((-lambertw(-3*x/2*exp(3*x/2))/(3*x/2*exp(3*x/2)))^(2/3)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((sum(k=0, N, (k+1)^(k-1)*(3*x/2*exp(3*x/2))^k/k!))^(2/3)))
  • PARI
    a(n) = sum(k=0, n, (3*k)^(n-k)*(3*k+2)^(k-1)*binomial(n, k))/2^(n-1);

Formula

E.g.f.: A(x) = exp( (-2/3) * LambertW(-3*x/2 * exp(3*x/2)) ).
E.g.f.: A(x) = ( -LambertW(-3*x/2 * exp(3*x/2)) / (3*x/2 * exp(3*x/2)) )^(2/3).
E.g.f.: A(x) = ( Sum_{k>=0} (k+1)^(k-1) * (3*x/2 * exp(3*x/2))^k / k! )^(2/3).
a(n) = (1/2^(n-1)) * Sum_{k=0..n} (3*k)^(n-k) * (3*k+2)^(k-1) * binomial(n,k).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (2^(n-1) * exp(n - 2/3) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Feb 17 2023

A161628 E.g.f.: A(x,y) = LambertW(x*y*exp(x))/(x*y*exp(x)), as a triangle of coefficients T(n,k) = [x^n*y^k/n! ] A(x,y), read by rows.

Original entry on oeis.org

1, 0, -1, 0, -2, 3, 0, -3, 18, -16, 0, -4, 72, -192, 125, 0, -5, 240, -1440, 2500, -1296, 0, -6, 720, -8640, 30000, -38880, 16807, 0, -7, 2016, -45360, 280000, -680400, 705894, -262144, 0, -8, 5376, -217728, 2240000, -9072000, 16941456, -14680064, 4782969
Offset: 0

Views

Author

Paul D. Hanna, Jun 15 2009, Jun 16 2009, Jun 17 2009

Keywords

Comments

The sum of row r of the triangle is (-1)^r (see A244119). - Stanislav Sykora, Jun 21 2014

Examples

			Triangle begins:
1;
0, -1;
0, -2, 3;
0, -3, 18, -16;
0, -4, 72, -192, 125;
0, -5, 240, -1440, 2500, -1296;
0, -6, 720, -8640, 30000, -38880, 16807;
0, -7, 2016, -45360, 280000, -680400, 705894, -262144;
0, -8, 5376, -217728, 2240000, -9072000, 16941456, -14680064, 4782969;
0, -9, 13824, -979776, 16128000, -102060000, 304946208, -462422016, 344373768, -100000000; ...

Links

Crossrefs

Cf. A161552, A244119, A273954.

Programs

  • Maple
    A161628 := (n, k) -> (-1)^k*binomial(n, k)*(k+1)^(k-1)*k^(n-k):
seq(seq(A161628(n,k), k=0..n), n=0..8); # Peter Luschny, Jan 29 2023
  • Mathematica
    Join[{1}, Table[(-1)^k*Binomial[n, k]*(k + 1)^(k - 1)*k^(n - k), {n, 1, 10}, {k, 0, n}]] // Flatten (* G. C. Greubel, Nov 09 2017 *)
  • PARI
    {T(n,k)=(-1)^k*binomial(n,k)*(k+1)^(k-1)*k^(n-k)}
  • PARI
    {T(n,k)=local(A,LW=serreverse(x*exp(x+x*O(x^n))));A=subst(LW/x,x,x*y*exp(x));n!*polcoeff(polcoeff(A,n,x),k,y)}
  • PARI
    {T(n,k)=local(G=1+x);for(i=0,n,G=exp(x*y*exp(x*G+O(x^n))));n!*polcoeff(polcoeff(serreverse(x*G)/x,n,x),k,y)}

Formula

T(n,k) = (-1)^k*C(n,k)*(k+1)^(k-1)*k^(n-k).
E.g.f. satisfies: A(x,y) = exp(-x*y*exp(x)*A(x,y)).
E.g.f.: A(x,y) = Sum_{n>=0} (n+1)^(n-1) * (-x)^n*y^n*exp(n*x)/n!.
E.g.f.: A(x,y) = (1/x)*Series_Reversion[x*G(x,y)] where G(x,y) = exp(x*y*exp(x*G(x,y))) is the e.g.f. of A161552.
More generally, if G(x,y) = exp(p*x*y*exp(q*x)*G(x,y)),
where G(x,y)^m = Sum_{n>=0} g(n,m)*x^n/n!,
then g(n,m) = C(n,k)*p^k*q^(n-k) * m*(k+m)^(k-1) * k^(n-k)
and G(x,y) = LambertW(-p*x*y*exp(q*x))/(-p*x*y*exp(q*x)).

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

A362654 E.g.f. satisfies A(x) = exp( x * exp(x^2) * A(x) ).

Original entry on oeis.org

1, 1, 3, 22, 197, 2316, 33967, 595624, 12190761, 285479056, 7531645211, 221124649824, 7152276636397, 252742471065280, 9688895208298503, 400510408002257536, 17759663471017945553, 840937887639033467136, 42351198256293556043827
Offset: 0

Views

Author

Seiichi Manyama, Apr 28 2023

Keywords

Links

Crossrefs

Cf. A273954, A362655.
Cf. A362653, A362673.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*exp(x^2)))))

Formula

E.g.f.: exp( -LambertW(-x * exp(x^2)) ).
a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)^k * (n-2*k+1)^(n-2*k-1) / (k! * (n-2*k)!).
a(n) ~ sqrt(1 + LambertW(2*exp(-2))) * 2^(n/2) * n^(n-1) / (exp(n-1) * LambertW(2*exp(-2))^(n/2)). - Vaclav Kotesovec, Aug 05 2025

A362656 E.g.f. satisfies A(x) = exp( x * exp(x) * A(x)^3 ).

Original entry on oeis.org

1, 1, 9, 145, 3569, 119041, 5025145, 256991953, 15448193633, 1067634195841, 83414064659561, 7270683884044945, 699503964027087697, 73631519384051331457, 8417768844410686595801, 1038658083084399115865041, 137579671405398060549801665
Offset: 0

Views

Author

Seiichi Manyama, Apr 28 2023

Keywords

Links

Crossrefs

Cf. A273954, A360473, A362671, A362672.
Cf. A052752.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-3*x*exp(x))/3)))

Formula

E.g.f.: exp( -LambertW(-3*x * exp(x))/3 ).
a(n) = Sum_{k=0..n} k^(n-k) * (3*k+1)^(k-1) * binomial(n,k).

A360176 Triangle read by rows. T(n, k) = Sum_{j=k..n} binomial(n, j) * (-j)^(n - j) * (-1)^(j - k)* A360177(j, k).

Original entry on oeis.org

1, 0, 1, 0, -5, 1, 0, 37, -15, 1, 0, -393, 223, -30, 1, 0, 5481, -3815, 745, -50, 1, 0, -95053, 76051, -18870, 1865, -75, 1, 0, 1975821, -1749811, 514381, -65730, 3920, -105, 1, 0, -47939601, 45876335, -15316854, 2358181, -183610, 7322, -140, 1
Offset: 0

Views

Author

Peter Luschny, Jan 28 2023

Keywords

Examples

			Triangle T(n, k) starts:
[0] 1;
[1] 0,         1;
[2] 0,        -5,        1;
[3] 0,        37,      -15,         1;
[4] 0,      -393,      223,       -30,       1;
[5] 0,      5481,    -3815,       745,     -50,       1;
[6] 0,    -95053,    76051,    -18870,    1865,     -75,    1;
[7] 0,   1975821, -1749811,    514381,  -65730,    3920, -105,    1;
[8] 0, -47939601, 45876335, -15316854, 2358181, -183610, 7322, -140, 1;

Crossrefs

Cf. A360177, A273954 (column 1), A028895 (subdiagonal).

Programs

  • Maple
    T := (n, k) -> add(binomial(n, j) * (-j)^(n - j) * (-1)^(j - k) * A360177(j, k), j = k..n): for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
# Alternative:
egf := k -> (1 - exp(-LambertW(x*exp(-x))))^k / k!:
ser := k -> series(egf(k), x, 22): T := (n, k) -> n!*coeff(ser(k), x, n):
for n from 0 to 8 do seq(T(n, k), k = 0..n) od;

Formula

E.g.f. of column k: (1 - exp(-LambertW(x*exp(-x))))^k / k!.
Showing 1-10 of 19 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.