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 1-3 of 3 results.

A078945 Row sums of A078939.

Original entry on oeis.org

1, 5, 29, 189, 1357, 10589, 88909, 797085, 7583373, 76179037, 804638925, 8904557341, 102929260813, 1239432543709, 15511264432973, 201330839371421, 2705249923950477, 37567754666530141, 538369104335121869
Offset: 0

Views

Author

Paul D. Hanna, Dec 18 2002

Keywords

Comments

Equals A078944(n+1)/4.

Links

Crossrefs

Column k=4 of A335975.
Cf. A078939, A078944, A000110, A035009, A078940.

Programs

  • Maple
    A078945 := proc(n) local a,b,i;
a := [seq(2,i=1..n)]; b := [seq(1,i=1..n)];
exp(-x)*hypergeom(a,b,x); round(evalf(subs(x=4,%),66)) end:
seq(A078945(n),n=0..18); # Peter Luschny, Mar 30 2011
  • Mathematica
    Table[n!, {n, 0, 20}]CoefficientList[Series[E^(4E^x-4+x), {x, 0, 20}], x]
Table[1/E^4/4*Sum[m^n/m!*4^m,{m,0,Infinity}],{n,1,20}] (* Vaclav Kotesovec, Mar 12 2014 *)
Table[BellB[n+1, 4]/4, {n, 0, 20}] (* Vaclav Kotesovec, Jun 26 2022 *)

Formula

E.g.f.: exp(4*(exp(x)-1)+x).
Stirling transform of [1, 4, 4^2, 4^3, ...]. - Gerald McGarvey, Jun 01 2005
Define f_1(x), f_2(x), ... such that f_1(x)=e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n)=e^{-4}*f_n(4). - Milan Janjic, May 30 2008
G.f.: 1/(Q(0) - 4*x) where Q(k) = 1 - x*(k+1)/( 1 - 4*x/Q(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 22 2013
G.f.: T(0)/(1-5*x), where T(k) = 1 - 4*x^2*(k+1)/( 4*x^2*(k+1) - (1-5*x-x*k)*(1-6*x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 28 2013
a(n) = exp(-4) * Sum_{k>=0} (k + 1)^n * 4^k / k!. - Ilya Gutkovskiy, Apr 20 2020
a(n) ~ n^(n+1) * exp(n/LambertW(n/4) - n - 4) / (4 * sqrt(1 + LambertW(n/4)) * LambertW(n/4)^(n+1)). - Vaclav Kotesovec, Jun 26 2022
a(0) = 1; a(n) = a(n-1) + 4 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Dec 05 2023

Extensions

More terms from Robert G. Wilson v, Dec 19 2002

A078940 Row sums of A078938.

Original entry on oeis.org

1, 4, 19, 103, 622, 4117, 29521, 227290, 1865881, 16239523, 149142952, 1439618143, 14555631781, 153700654036, 1690684883191, 19328770917499, 229203640111870, 2814018686591089, 35711716110387589, 467766675528462562
Offset: 0

Views

Author

Paul D. Hanna, Dec 18 2002

Keywords

Comments

Divide by 3^n and insert an initial 1 to get sequence that shifts left one place under 1/3 order binomial transformation. - Franklin T. Adams-Watters, Jul 13 2006
Binomial transform of A027710. - Vaclav Kotesovec, Jun 26 2022

Links

Crossrefs

Column k=3 of A335975.
Cf. A027710, A035009, A078938, A078945, A355254.

Programs

  • Maple
    A078940 := proc(n) local a,b,i;
a := [seq(2,i=1..n)]; b := [seq(1,i=1..n)];
exp(-x)*hypergeom(a,b,x); round(evalf(subs(x=3,%),66)) end:
seq(A078940(n),n=0..19); # Peter Luschny, Mar 30 2011
  • Mathematica
    Table[n!, {n, 0, 20}]CoefficientList[Series[E^(3E^x-3+x), {x, 0, 20}], x]
Table[1/E^3/3*Sum[m^n/m!*3^m,{m,0,Infinity}],{n,1,20}] (* Vaclav Kotesovec, Mar 12 2014 *)
Table[BellB[n+1, 3]/3, {n, 0, 20}] (* Vaclav Kotesovec, Jan 15 2016 *)
nmax = 20; Clear[g]; g[nmax+1] = 1; g[k_] := g[k] = 1 - (k+4)*x - 3*(k+1)*x^2/g[k+1]; CoefficientList[Series[1/g[0], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 15 2016, after Sergei N. Gladkovskii *)

Formula

E.g.f.: exp(3*(exp(x)-1)+x).
Stirling transform of [1, 3, 3^2, 3^3, ...]. - Gerald McGarvey, Jun 01 2005
Define f_1(x), f_2(x), ... such that f_1(x)=e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n)=e^{-3}*f_n(3). - Milan Janjic, May 30 2008
G.f.: 1/T(0), where T(k) = 1 - (k+4)*x - 3*(k+1)*x^2/T(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2016
a(n) = exp(-3) * Sum_{k>=0} (k + 1)^n * 3^k / k!. - Ilya Gutkovskiy, Apr 20 2020
a(n) ~ n^(n+1) * exp(n/LambertW(n/3) - n - 3) / (3 * sqrt(1 + LambertW(n/3)) * LambertW(n/3)^(n+1)). - Vaclav Kotesovec, Jun 26 2022
a(0) = 1; a(n) = a(n-1) + 3 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Dec 05 2023

Extensions

More terms from Robert G. Wilson v, Dec 19 2002

A335977 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(1 - exp(x)) + x).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -1, -1, 1, 1, -3, 1, 3, 2, 1, 1, -4, 5, 7, 7, 9, 1, 1, -5, 11, 5, -8, -13, 9, 1, 1, -6, 19, -9, -43, -65, -89, -50, 1, 1, -7, 29, -41, -74, -27, 37, -45, -267, 1, 1, -8, 41, -97, -53, 221, 597, 1024, 1191, -413, 1, 1, -9, 55, -183, 92, 679, 961, 805, 1351, 4723, 2180, 1
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2020

Keywords

Examples

			Square array begins:
  1,  1,   1,   1,   1,   1,    1, ...
  1,  0,  -1,  -2,  -3,  -4,   -5, ...
  1, -1,  -1,   1,   5,  11,   19, ...
  1, -1,   3,   7,   5,  -9,  -41, ...
  1,  2,   7,  -8, -43, -74,  -53, ...
  1,  9, -13, -65, -27, 221,  679, ...
  1,  9, -89,  37, 597, 961, -341, ...

Links

Crossrefs

Columns k=0-4 give: A000012, A293037, A309775, A320432, A320433.
Main diagonal gives A334241.
Cf. A292861, A335975.

Programs

  • Mathematica
    T[0, k_] := 1; T[n_, k_] := T[n - 1, k] - k * Sum[T[j, k] * Binomial[n - 1, j], {j, 0, n - 1}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Jul 03 2020 *)

Formula

T(0,k) = 1 and T(n,k) = T(n-1,k) - k * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.
T(n,k) = exp(k) * Sum_{j>=0} (j + 1)^n * (-k)^j / j!.
Showing 1-3 of 3 results.

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