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.

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A348087 a(n) = [x^n] Product_{k=1..n} 1/(1 - (2*k-1) * x).

Original entry on oeis.org

1, 1, 13, 330, 12411, 618870, 38461522, 2863440580, 248440887123, 24616763946918, 2742625188929990, 339386813915985836, 46184075261030623710, 6854605372617955658940, 1101943692701420653738500, 190748265085183804327197000, 35373318817392757170821576835
Offset: 0

Views

Author

Seiichi Manyama, Sep 28 2021

Keywords

Links

Crossrefs

Cf. A001147, A007820, A039755, A348085, A348088.

Programs

  • PARI
    a(n) = polcoef(1/prod(k=1, n, 1-(2*k-1)*x+x*O(x^n)), n);
  • PARI
    a(n) = if(n==0, 1, sum(k=0, n-1, (-1)^k*(2*k+1)^(2*n-1)*binomial(n-1, k))/((-2)^(n-1)*(n-1)!));

Formula

a(n) = A039755(2*n-1,n-1) for n > 0.
a(n) = (1/((-2)^(n-1) * (n-1)!)) * Sum_{k=0..n-1} (-1)^k * (2*k+1)^(2*n-1) * binomial(n-1,k) for n > 0.
a(n) ~ 2^(3*n - 1) * n^(n - 1/2) / (sqrt(Pi*(1-c)) * (2-c)^n * c^(n - 1/2) * exp(n)), where c = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599... - Vaclav Kotesovec, Oct 02 2021
From Seiichi Manyama, May 16 2025: (Start)
a(n) = Sum_{k=0..n} 2^k * binomial(2*n-1,k+n-1) * Stirling2(k+n-1,n-1) for n > 0.
a(n) = Sum_{k=0..n} (-2)^k * (2*n-1)^(n-k) * binomial(2*n-1,k+n-1) * Stirling2(k+n-1,n-1) for n > 0. (End)

A166973 Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m = 0.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 43, 18, 1, 1, 259, 241, 34, 1, 1, 1555, 2910, 785, 55, 1, 1, 9331, 33565, 15470, 1940, 81, 1, 1, 55987, 378546, 281085, 56210, 4046, 112, 1, 1, 335923, 4219993, 4875906, 1461495, 161406, 7518, 148, 1, 1, 2015539, 46755846, 82234489
Offset: 1

Views

Author

Roger L. Bagula, Oct 26 2009

Keywords

Comments

The recursion T(n, k) = (m*n - m*k + 1)*T(n-1, k-1) + (5*k - 4)*(m*k - (m - 1))*T(n-1, k) was intended to range over m values 0 to 4 as given by the original Mathematica code. This sequences is the case for m = 0. - G. C. Greubel, May 29 2016
With offset 0 in the rows and columns this is the Sheffer triangle S2[5,1] = (exp(x), (exp(5*x) - 1)/5). See S2[4,1] = A111578 (with offsets 0), S[3,1] = A111577 (with offsets 0), S2[2,1] = A039755

Examples

			Triangle T(n, k) starts:
n\k   1       2        3        4        5       6      7     8   9 10 ...
1:    1
2:    1       1
3:    1       7        1
4:    1      43       18        1
5:    1     259      241       34        1
6:    1    1555     2910      785       55       1
7:    1    9331    33565    15470     1940      81      1
8:    1   55987   378546   281085    56210    4046    112     1
9:    1  335923  4219993  4875906  1461495  161406   7518   148   1
10:   1 2015539 46755846 82234489 35567301 5658051 394464 12846 189  1
... Reformatted, - _Wolfdieter Lang_, Aug 13 2017

Links

Crossrefs

Cf. A111577.
S2[4,1] = A111578 (with offsets 0), S2[3,1] = A111577 (with offsets 0), S2[2,1] = A039755. - Wolfdieter Lang, Aug 13 2017

Programs

  • Mathematica
    A[n_, 1] := 1; A[n_, n_] := 1; A[n_, k_] := A[n - 1, k - 1] + (5*k - 4)*A[n - 1,k]; Flatten[ Table[A[n, k], {n, 10}, {k, n}]] (* modified by G. C. Greubel, May 29 2016 *)

Formula

T(n, k) = T(n - 1, k - 1) + (5*k - 4)*T(n - 1, k).
E.g.f. column k: int(exp(x)*((exp(5*x)-1)/5)^(k-1)/(k-1)!, x) + (-1)^k/A008548(k). - Wolfdieter Lang, Aug 13 2017

A225480 a(n) = B2(n) * C(n) where B2(n) are generalized Bernoulli numbers and C(n) the Clausen numbers.

Original entry on oeis.org

1, 0, -2, 0, 14, 0, -62, 0, 254, 0, -5110, 0, 2828954, 0, -114674, 0, 237036478, 0, -11499383114, 0, 183092554714, 0, -3584085584926, 0, 3965530936622474, 0, -573989008898786, 0, 6375197353574922166, 0, -9251189109760413581110, 0, 33111281730973040956798, 0
Offset: 0

Views

Author

Peter Luschny, May 30 2013

Keywords

Comments

The Clausen numbers C(n) are T(n, 1) in A160014.

Examples

			The numerators of 1/1, 0/2, -2/6, 0/2, 14/30, 0/2, -62/42, 0/2, 254/30, 0/2, -5110/66, 0/2, 2828954/2730, ... (the denominators are the Clausen numbers).

Links

Crossrefs

Cf. A141056, A160014.

Programs

  • Maple
    B := (n, m) -> add(add(add(((-1)^(n-v)/(j+1))*binomial(n,k)*binomial(j, v)*(m*v)^k, v = 0..j), j = 0..k), k = 0..n);
C := proc(n) numtheory[divisors](n);map(i->i+1,%);select(isprime,%);mul(i,i=%) end:
A225480 := n -> B(n, 2)*C(n); seq(A225480(n), n = 0..33);
  • Mathematica
    B[n_, m_] := Sum[((-1)^(n - v)/(j + 1))*Binomial[n, k]*Binomial[j, v]*If[k == 0, 1, (m*v)^k], {k, 0, n}, {j, 0, k}, {v, 0, j}];
c[n_] := Denominator[Sum[Boole[PrimeQ[d + 1]]/(d + 1), {d, Divisors[n]}]];
a[n_] := B[n, 2]*c[n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Aug 02 2019, from Maple *)
  • Sage
    @CachedFunction
def EulerianNumber(n, k, m) :   # The Eulerian numbers
    if n == 0: return 1 if k == 0 else 0
    return (m*(n-k)+m-1)*EulerianNumber(n-1,k-1,m)+(m*k+1)*EulerianNumber(n-1,k,m)
@CachedFunction
def B(n, m):   # The generalized Bernoulli numbers
    return add(add(EulerianNumber(n, j, m)*binomial(j, n - k)
           for j in (0..n))*(-1)^k/(k+1) for k in (0..n))
def A225480(n):
    if n == 0: return 1
    C = mul(filter(lambda s: is_prime(s) , map(lambda i: i+1, divisors(n))))
    return C*B(n, 2)
print([A225480(n) for n in (0..33)])

Formula

Let B(n,m) = sum_{k = 0..n} sum_{j = 0..k} sum_{v = 0..j} ((-1)^(n-v)/(j+1)) *binomial(n,k)*binomial(j,v)*(m*v)^k then a(n) = B(n,2)*A141056(n).
Let B2(n) = sum_{k=0..n} ((-1)^k*k!/(k+1)) S_{2}(n, k) where S_{2}(n, k) the Stirling-Frobenius subset numbers A039755(n, k) then a(n) = B2(n)*A141056(n).
Previous Showing 31-33 of 33 results.

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