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 1-5 of 5 results.

A136658 Row sums of unsigned triangle A136656 and also of triangle 2*A136657.

Original entry on oeis.org

1, 2, 10, 68, 580, 5912, 69784, 933200, 13912336, 228390560, 4088594464, 79186453568, 1648396356160, 36678170613632, 868239454798720, 21776352497954048, 576629116655862016, 16069766602389885440, 470015788927133039104, 14392014594072635786240
Offset: 0

Views

Author

Wolfdieter Lang, Feb 22 2008

Keywords

Links

Crossrefs

Cf. A049376, A136656, A136657.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+1)!*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017
  • Mathematica
    a[n_] := Sum[ StirlingS1[n, k] * BellB[k] * (-1)^(n-k) * 2^k, {k, 0, n}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 09 2013, after Paul D. Hanna *)
Table[Sum[BellY[n, k, (Range[n] + 1)!], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)
  • PARI
    {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
{a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k) * (-1)^(n-k)*2^k)}
/* Paul D. Hanna, Dec 25 2011 */

Formula

a(n) = Sum_{k=0..n} (-1)^n*A136656(n,k), n>=0.
E.g.f.: exp(x*(2-x)/(1-x)^2) (from Jabotinsky type triangle).
a(n) = Sum_{k=0..n} Stirling1(n, k) * Bell(k) * (-1)^(n-k) * 2^k. - Paul D. Hanna, Dec 25 2011
a(n) = (3*n-1)*a(n-1) - 3*(n-2)*(n-1)*a(n-2) + (n-3)*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Sep 25 2013
a(n) ~ 2^(1/6)*n^(n-1/6) * exp((n/2)^(1/3)+3*(n/2)^(2/3)-n-2/3) / sqrt(3) * (1 + 7/(27*(n/2)^(1/3)) - 422/(3645*(n/2)^(2/3))). - Vaclav Kotesovec, Sep 25 2013
Representation as special values of hypergeometric functions 2F2, in Maple notation: a(n) = (n+1)!*hypergeom([(1/2)*n+1, (1/2)*n+3/2], [3/2, 2], 1)*exp(-1), n = 1,2,... . - Karol A. Penson, Jul 28 2018

A136656 Coefficients for rewriting generalized falling factorials into ordinary falling factorials.

Original entry on oeis.org

1, 0, -2, 0, 6, 4, 0, -24, -36, -8, 0, 120, 300, 144, 16, 0, -720, -2640, -2040, -480, -32, 0, 5040, 25200, 27720, 10320, 1440, 64, 0, -40320, -262080, -383040, -199920, -43680, -4032, -128, 0, 362880, 2963520, 5503680, 3764880, 1142400, 163968, 10752, 256, 0, -3628800, -36288000, -82978560
Offset: 0

Views

Author

Wolfdieter Lang, Feb 22 2008, Sep 09 2008

Keywords

Comments

Generalization of (signed) Lah number triangle A008297 (amended with a trivial row n=0 and a column k=0 in order to have a Sheffer triangle structure of the Jabotinsky type).
product(s*t-j,j=0..n-1) := fallfac(s*t,n) (falling factorial with n factors) is called generalized factorial of t of order n and scale parameter s in the Charalambides reference p. 301 ch. 8.4.
The s-family of triangles L(s;n,k) (in the Charalambides reference called C(n,k;-s)) is defined for integer s by fallfac(-s*t,n) = ((-1)^n)*risefac(s*t,n) = sum(L(s;n,k)*fallfac(t,k),k=0..n), n>=0. risefac(x,n):=product(x+j,j=0..n-1) for the rising factorials.
For positive s the signless triangles |L(s;n,k)| = L(s;n,k)*(-1)^n satisfies risefac(s*t,n) = sum(|L(s;n,k)|*fallfac(t,k),k=0..n), n>=0.
For negative s see the combinatorial interpretation given in the Charalambides reference, Example 8.8, p. 313: Coupon collector's problem.
|T(n,k)| = B_{n,k}((j+2)!; j>=0) where B_{n,k} are the partial Bell polynomials. - Peter Luschny, May 11 2015

Examples

			Triangle starts:
[1]
[0,   -2]
[0,    6,    4]
[0,  -24,  -36,   -8]
[0,  120,  300,  144,  16]
...
Recurrence: a(4,2) = -7*a(3,2)-2*a(3,1) = -7*(-36) -2*(-24) = 300.
a(4,2)=300 as sum over the M3 numbers A036040 for the 2 parts partitions of 4: 4*fallfac(-2,1)^1*fallfac(-2,3)^1 + 3*fallfac(-2,2)^2 = 4*(-2)*(-24)+3*6^2 = 300.
Row n=3: [0,-24,-36,-8] for the coefficients in rewriting fallfac(-2*t,3)=((-1)^3)*risefac(2*t,3) = ((-1)^3)*(2*t)*(2*t+1)*(2*t+2) = 0*1 -24*t -36*t*(t-1) -8*t*(t-1)*(t-2).

References

  • Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, ch. 8.4 p. 301 ff. Table 8.3 (with row n=0 and column k=0 and s=-2).

Links

Crossrefs

Column sequences (unsigned) 2*A001710, 4*A136659, 8*A136660, 16*A136661 for k=1..4.
Cf. A136657 without row n=0 and column k=0, divided by 2.

Programs

  • Maple
    # The function BellMatrix is defined in A264428.
BellMatrix(n -> (-1)^(n+1)*(n+2)!, 9); # Peter Luschny, Jan 27 2016
  • Mathematica
    fallfac[n_, k_] := Pochhammer[n - k + 1, k]; a[n_, k_] := Sum[(-1)^(k - r)*Binomial[k, r]*fallfac[-2*r, n], {r, 0, k}]/k!; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 09 2013 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[(-1)^(# + 1)*(# + 2)!&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (*Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
  • Sage
    @CachedFunction
def T(n, k):  # unsigned case
    if k == 0: return 1 if n == 0 else 0
    return sum(T(n-j-1,k-1)*(j+1)*(j+2)*gamma(n)/gamma(n-j) for j in (0..n-k))
for n in range(7): [T(n,k) for k in (0..n)] # Peter Luschny, Mar 31 2015

Formula

Recurrence: a(n,k) = 0 if n
E.g.f. column k: (1/(1+x)^2 - 1)^k/k!, k>=0. From the Charalambides reference Theorem 8.14, p. 305 for s=-2.(hence a Sheffer triangle of Jabotinsky type).
a(n,k) = sum(((-1)^(k-r))*binomial(k,r)*fallfac(-2*r,n),r=0..k)/k!, n>=k>=0. From the Charalambides reference Theorem 8.15, p. 306 for s=-2.
a(n,k) = sum(S1(n,r)*S2(r,k)*(-2)^r,r=k..n) with the Stirling numbers S1(n,r)= A048993(n,r) and S2(r,k)= A048993(r,k). From the Charalambides reference Theorem 8.13, p.304 for s=-2.
a(n,k) = sum(M_3(n,k,q)*product(fallfac(-2,j)^e(n,m,q,j),j=1..n),q=1..p(n,k)) if n>=k>=1, else 0. Here p(n,k)=A008284(n,m), the number of k parts partitions of n, the M_3 partition numbers are given in A036040 and e(n,m,q,j) is the exponent of j in the q-th k parts partition of n. Rewritten eq. (8.50), Theorem 8.16, p. 307, from the Charalambides reference for s=-2.
Recurrence for the unsigned case: a(n,k) = Sum_{j=0..n-k} a(n-j-1,k-1)*C(n-1,j)*(j+2)! if k<>0 else k^n. - Peter Luschny, Mar 31 2015

A136659 Unsigned third column (k=2) of triangle A136656 divided by 4.

Original entry on oeis.org

1, 9, 75, 660, 6300, 65520, 740880, 9072000, 119750400, 1696464000, 25686460800, 414096883200, 7083236160000, 128152088064000, 2445351068160000, 49084865077248000, 1033983353475072000, 22808456326656000000, 525810946517176320000, 12645008187498086400000
Offset: 0

Views

Author

Wolfdieter Lang, Feb 22 2008

Keywords

Comments

Also unsigned second column of triangle A136657 divided by 2.

References

  • Charalambos A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, Table 8.3, p. 311, with s=-2, k=2 column/4.

Crossrefs

Cf. A001710 (1/2 of unsigned k=1 column of A136657). A136660 (k=3 column divided by 8), A136656.
Cf. A001113, A001620, A091725, A099285.

Programs

  • Mathematica
    a[n_] := (n + 8)*(n + 1)*(n + 3)!/48; Array[a, 20, 0] (* Amiram Eldar, Aug 31 2025 *)

Formula

a(n) = |A136656(n+2,2)|/4, n>=0.
E.g.f.: (2+6*x-3*x^2)/(2*(1-x)^6) (derived from the one given for the column k=2 under A136656).
a(n) = (n+4)!/2 * sum((k+1)!/(k+4)!,k=1..n), with offset 1. - Gary Detlefs, Jul 27 2010
a(n) = (1/48) * (n+8)*(n+1)*(n+3)!. - Gary Detlefs, Aug 03 2010
From Amiram Eldar, Aug 31 2025: (Start)
Sum_{n>=0} 1/a(n) = 44836/245 - 480*e/7 - 24*gamma/7 + 24*ExpIntegralEi(1)/7, where e = A001113, gamma = A001620, and ExpIntegralEi(1) = A091725.
Sum_{n>=0} (-1)^n/a(n) = 39724/245 - 3120/(7*e) + 24*gamma/7 - 24*ExpIntegralEi(-1)/7, where ExpIntegralEi(-1) = -A099285. (End)

A136660 Unsigned fourth column (k=3) of triangle A136656 divided by 8.

Original entry on oeis.org

1, 18, 255, 3465, 47880, 687960, 10372320, 164656800, 2754259200, 48518870400, 899026128000, 17495593315200, 356995102464000, 7625049239808000, 170196434343936000, 3963602854987776000, 96160451873181696000
Offset: 0

Views

Author

Wolfdieter Lang, Feb 22 2008

Keywords

Comments

Also unsigned third column of triangle A136657 divided by 4.

References

  • Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, Table 8.3, p. 311, with s=-2, k=3 column/8.

Crossrefs

Cf. A136656, A136659 (column k=2), A136661 (column k=4).

Formula

a(n) = |A136656(n+3,3)|/8, n>=0.
E.g.f.: (2+18*x+3*x^2-12*x^3+3*x^4)/(2*(1-x)^9) (derived from the one given for the column k=3 under A136656).

A136661 Unsigned fifth column (k=4) of triangle A136656 divided by 16.

Original entry on oeis.org

1, 30, 645, 12495, 235305, 4452840, 86070600, 1713927600, 35318883600, 754896542400, 16751853518400, 386036370720000, 9235831629024000, 229285008336384000, 5902321642753536000, 157423965566579712000
Offset: 0

Views

Author

Wolfdieter Lang, Feb 22 2008

Keywords

Comments

Also unsigned fourth column of triangle A136657 divided by 8.

References

  • Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, Table 8.3, p. 311, with s=-2, k=4 column/16.

Crossrefs

Cf. A136656, A136660 (column k=3).

Formula

a(n) = |A136656(n+4,4)|/16, n>=0.
E.g.f.: (8+144*x+228*x^2-220*x^3-45*x^4+60*x^5-10*x^6)/(8*(1-x)^12) (derived from the one given for the column k=4 under A136656).
Showing 1-5 of 5 results.

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