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 1-4 of 4 results.

A345652 Expansion of the e.g.f. exp(-1 + (x + 1)*exp(-x)).

Original entry on oeis.org

1, 0, -1, 2, 0, -16, 65, -78, -749, 6232, -22068, -28920, 1004685, -7408740, 22263215, 157632230, -2874256740, 21590948480, -53087332675, -956539294506, 16344490525835, -132605481091060, 294656170409328, 9113173803517344, -167298122286332823
Offset: 0

Views

Author

Mélika Tebni, Jun 21 2021

Keywords

Comments

For all p prime, a(p)/(p-1) == 1 (mod p). - Mélika Tebni, Mar 21 2022

Examples

			exp(-1+(x+1)*exp(-x)) = 1 - x^2/2! + 2*x^3/3! - 16*x^5/5! + 65*x^6/6! - 78*x^7/7! - 749*x^8/8! + 6232*x^9/9! + ...

Links

Crossrefs

Cf. A003725, A014182, A327006.
Cf. A292935 (without 1+x: EGF e^(e^(-x)-1)), A000110 (absolute values: Bell numbers, EGF e^(e^x-1))
Cf. A003725, A106828, A347210, A347571.

Programs

  • Maple
    a := series(exp(-1+(x+1)*exp(-x)), x=0, 25): seq(n!*coeff(a, x, n), n=0..24);
a := proc(n) option remember; `if`(n=0, 1, add((n-1)*binomial(n-2, k)*(-1)^(n-1-k)*a(k), k=0..n-2)) end: seq(a(n), n=0..24);
# third program:
A345652 := n -> add((-1)^(n-k)*combinat[bell](k)*A106828(n, k), k=0..iquo(n, 2)):
seq(A345652(n), n=0..24); # Mélika Tebni, Sep 21 2021
  • Mathematica
    nmax = 24; CoefficientList[Series[Exp[-1+(x+1)*Exp[-x]], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    seq(n) = {Vec(serlaplace(exp(-1+(x+1)*exp(-x + O(x*x^n)))))} \\ Andrew Howroyd, Jun 21 2021
  • PARI
    a(n) = if(n==0, 1, sum(k=2, n, (-1)^(k-1)*(k-1)*binomial(n-1, k-1)*a(n-k))); \\ Seiichi Manyama, Mar 15 2022

Formula

The e.g.f. y(x) satisfies y' = -x*y*exp(-x).
a(n) = Sum_{k=0..n-2} (n-1)*binomial(n-2, k)*(-1)^(n-1-k)*a(k) for n > 0.
Conjecture: a(n) = 0 for only n = 1 and n = 4.
Conjecture: For all p prime, a(p)^2 == 1 (mod p).
Stronger conjecture: For n > 1, a(n) == -1 (mod n) iff n is a prime or 6. - M. F. Hasler, Jun 23 2021
a(n) = Sum_{k=0..floor(n/2)} (-1)^(n-k)*Bell(k)*A106828(n, k). - Mélika Tebni, Sep 21 2021
a(n) = Sum_{k=0..n} (-1)^k*A003725(n-k)*Bell(k)*binomial(n, k). - Mélika Tebni, Mar 21 2022

A347571 Expansion of the e.g.f. (-1 - 2*x - 2*log(1 - x) + exp(-2*x) / (1 - x)^2) / 4 + 1.

Original entry on oeis.org

1, 0, 1, 2, 9, 44, 280, 2064, 17528, 167488, 1777536, 20721920, 263055232, 3610443264, 53256280064, 839974309888, 14103897738240, 251146689069056, 4726795773018112, 93746994502828032, 1954053073794596864, 42702893781890498560, 976276451410488066048, 23303485413254033309696
Offset: 0

Views

Author

Mélika Tebni, Sep 07 2021

Keywords

Comments

For all p prime, a(p) == -1 (mod p).
For n > 1, a(n) == 0 (mod (n-1)).

Examples

			E.g.f.: 1 + x^2/2! + 2*x^3/3! + 9*x^4/4! + 44*x^5/5! + 280*x^6/6! + 2064*x^7/7! + 17528*x^8/8! + 167488*x^9/9! + ...
a(11) = Sum_{k=0..5} ceiling(2^(k-2))*A106828(11, k) = 20721920.
For k = 0, A106828(11,0) = 0.
For k = 1, ceiling(2^(1-2))*A106828(11, 1) == -1 (mod 11), because ceiling(2^(1-2)) = 1 and A106828(11, 1) = (11-1)!
For k >= 2, ceiling(2^(k-2))*A106828(11, k) == 0 (mod 11), because A106828(11, k) == 0 (mod 11), result a(11) == -1 (mod 11).
a(10) = Sum_{k=0..5} ceiling(2^(k-2))*A106828(10, k) = 1777536.
a(10) == 0 (mod (10-1)), because for k >= 0, A106828(10, k) == 0 (mod 9).

Crossrefs

Cf. A106828, A343482, A345697, A345969, A346119, A347210.

Programs

  • Maple
    a := series((-1-2*x-2*log(1-x)+exp(-2*x)/(1-x)^2)/4+1, x=0, 24):
seq(n!*coeff(a, x, n), n=0..23);
# second program:
a := n -> add(ceil(2^(k-2))*A106828(n, k), k=0..iquo(n, 2)):
seq(a(n), n=0..23);
  • Mathematica
    CoefficientList[Series[(-1 - 2*x - 2*Log[1 - x] + Exp[-2*x]/(1 - x)^2)/4 + 1, {x, 0, 23}], x]*Range[0, 23]!
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace((-1-2*x-2*log(1-x)+exp(-2*x)/(1-x)^2)/4 + 1)) \\ Michel Marcus, Sep 07 2021

Formula

a(n) = Sum_{k=0..floor(n/2)} ceiling(2^(k-2))*A106828(n, k).
a(n) ~ n * n! / (4*exp(2)). - Vaclav Kotesovec, Sep 10 2021

A348208 a(n) = Sum_{k=0..floor(n/2)} (-1)^(k-1)*(k-1)^2*A106828(n, k).

Original entry on oeis.org

-1, 0, 0, 0, -3, -20, -70, -84, 1267, 18824, 209484, 2284920, 26010369, 314864628, 4073158102, 56304102596, 830061867975, 13016975343184, 216535182535928, 3810394068301296, 70744547160678501, 1382375535029293500, 28364229790262962386, 609820072529413714012
Offset: 0

Views

Author

Mélika Tebni, Oct 07 2021

Keywords

Comments

For all p prime, a(p) == 0 (mod p*(p-1)).

Examples

			E.g.f.: -1 - 3*x^4/4! - 20*x^5/5! - 70*x^6/6! - 84*x^7/7! + 1267*x^8/8! + 18824*x^9/9! + ...
a(11) = Sum_{k=0..5} (-1)^(k-1)*(k-1)^2*A106828(11, k).
a(11) = (-1)*1*0 + (1)*0*3628800 + (-1)*1*6636960 + (1)*4*3678840 + (-1)*9*705320 + (1)*16*34650 = 2284920.
For k = 0, A106828(11,0) = 0.
For k = 1, (1-1)^2 = 0.
For 2 <= k <= 5, A106828(11, k) == 0 (mod 11*10).
Result a(11) == 0 (mod 11*10).

Crossrefs

Cf. A106828, A347210, A347571.

Programs

  • Maple
    a := series((-1+2*x-2*x^2+x^3+(1-x)*(log((1-x)^(1-2*x))-(log(1-x))^2))*exp(x), x=0, 24):
seq(n!*coeff(a, x, n), n=0..23);
# second program:
a := n -> add((-1)^(k-1)*(k-1)^2*A106828(n, k), k=0..iquo(n, 2)):
seq(a(n), n=0..23);
  • Mathematica
    CoefficientList[Series[(-1+2*x-2*x^2+x^3+(1-x)*(Log[(1-x)^(1-2*x)]-(Log[1-x])^2))*Exp[x], {x, 0, 23}], x]*Range[0, 23]!
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace((-1 + 2*x - 2*x^2 + x^3 + (1 - x)*(log((1 - x)^(1 - 2*x)) - (log(1 - x))^2))*exp(x))) \\ Michel Marcus, Oct 07 2021

Formula

E.g.f.: (-1 + 2*x - 2*x^2 + x^3 + (1 - x)*(log((1 - x)^(1 - 2*x)) - (log(1 - x))^2))*exp(x).
a(n) ~ 2 * exp(1) * log(n) * n! / n^2 * (1 + (gamma - 3/2)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 09 2021

A349959 a(n) = Sum_{k=0..floor(n/2)} (k-1)^2*A106828(n, k).

Original entry on oeis.org

1, 0, 0, 0, 3, 20, 190, 1764, 17773, 192632, 2250036, 28254600, 380304639, 5468906508, 83750505826, 1361579283596, 23431400945145, 425669127018416, 8142731710207432, 163636478165355408, 3447201944202849819, 75973975479088955460, 1748531872985454054246, 41951755708613404583732
Offset: 0

Views

Author

Mélika Tebni, Dec 07 2021

Keywords

Comments

For all p prime, a(p) == 0 (mod p*(p-1)).

Examples

			E.g.f.: 1 + 3*x^4/4! + 20*x^5/5! + 190*x^6/6! + 1764*x^7/7! + 17773*x^8/8! + 192632*x^9/9! + ...
a(13) = Sum_{k=0..6} (k-1)^2*A106828(13, k).
a(13) =  1*0 + 0*479001600 + 1*967524480 + 4*647536032 + 9*177331440 + 16*18858840 + 25*540540 = 5468906508.
For k = 0, A106828(13, 0) = 0.
For k = 1, (1-1)^2 = 0.
For 2 <= k <= 6, A106828(13, k) == 0 (mod 13*12).
Result a(13) == 0 (mod 13*12).

Crossrefs

Cf. A106828, A347210, A347571, A348208.

Programs

  • Maple
    a := n -> add((k-1)^2*A106828(n, k), k=0..iquo(n, 2)):
seq(a(n), n=0..23);
# second program:
a := series((-2-x+(3+log((1-x)^(1+2*x))+(log(1-x))^2)/(1-x))/exp(x), x=0, 24):
seq(n!*coeff(a, x, n), n=0..23);
  • Mathematica
    CoefficientList[Series[(-2-x+(3+Log[(1-x)^(1+2*x)]+(Log[1-x])^2)/(1-x))/Exp[x], {x, 0, 23}], x]*Range[0, 23]!
  • PARI
    E2(n, m) = sum(k=0, n-m, (-1)^(n+k)*binomial(2*n+1, k)*stirling(2*n-m-k+1, n-m-k+1, 1)); \\ A008517
ast1(n, k) = if ((n==0) && (k==0), 1, sum(j=0, n-k, binomial(j, n-2*k)*E2(n-k, j+1))); \\ A106828
a(n) = sum(k=0, n\2, (k-1)^2*ast1(n, k)); \\ Michel Marcus, Dec 07 2021

Formula

E.g.f.: (-2 - x + (3 + log((1 - x)^(1 + 2*x)) + (log(1 - x))^2) / (1 - x)) / exp(x).
a(n) ~ n! * exp(-1) * log(n)^2 * (1 + (2*gamma - 3)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 09 2021
Showing 1-4 of 4 results.

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

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