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.

User: Mark Shattuck

Mark Shattuck's wiki page.

Mark Shattuck has authored 2 sequences.

A203412 Triangle read by rows, a(n,k), n>=k>=1, which represent the s=3, h=1 case of a two-parameter generalization of Stirling numbers arising in conjunction with normal ordering.

Original entry on oeis.org

1, 1, 1, 4, 3, 1, 28, 19, 6, 1, 280, 180, 55, 10, 1, 3640, 2260, 675, 125, 15, 1, 58240, 35280, 10360, 1925, 245, 21, 1, 1106560, 658000, 190680, 35385, 4620, 434, 28, 1, 24344320, 14266560, 4090240, 756840, 100065, 9828, 714, 36, 1
Offset: 1

Views

Author

Mark Shattuck, Jan 01 2012

Keywords

Comments

Also the Bell transform of the triple factorial numbers A007559 which adds a first column (1,0,0 ...) on the left side of the triangle. For the definition of the Bell transform see A264428. See A051141 for the triple factorial numbers A032031 and A004747 for the triple factorial numbers A008544 as well as A039683 and A132062 for the case of double factorial numbers. - Peter Luschny, Dec 23 2015

Examples

			Triangle starts:
[    1]
[    1,     1]
[    4,     3,     1]
[   28,    19,     6,    1]
[  280,   180,    55,   10,   1]
[ 3640,  2260,   675,  125,  15,  1]
[58240, 35280, 10360, 1925, 245, 21, 1]

Links

Crossrefs

Cf. A007559, A032031, A039683, A051141, A132062, A264428.

Programs

  • Maple
    A203412 := (n,k) -> (n!*3^n)/(k!*2^k)*add((-1)^j*binomial(k,j)*binomial(n-2*j/3-1, n), j=0..k): seq(seq(A203412(n,k),k=1..n),n=1..9); # Peter Luschny, Dec 21 2015
  • Mathematica
    Table[(n! 3^n)/(k! 2^k) Sum[ (-1)^j Binomial[k, j] Binomial[n - 2 j/3 - 1, n], {j, 0, k}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Dec 23 2015 *)
  • Sage
    # uses[bell_transform from A264428]
triplefactorial = lambda n: prod(3*k + 1 for k in (0..n-1))
def A203412_row(n):
    trifact = [triplefactorial(k) for k in (0..n)]
    return bell_transform(n, trifact)
[A203412_row(n) for n in (0..8)] # Peter Luschny, Dec 21 2015

Formula

(1) Is given by the recurrence relation
a(n+1,k) = a(n,k-1)+(3*n-2*k)*a(n,k) if n>=0 and k>=1, along with the initial values a(n,0) = delta_{n,0} and a(0,k) = delta_{0,k} for all n,k>=0.
(2) Is given explicitly by
a(n,k) = (n!*3^n)/(k!*2^k)*Sum{j=0..k} (-1)^j*C(k,j)*C(n-2*j/3-1,n) for all n>=k>=1.
a(n,1) = A007559(n-1). - Peter Luschny, Dec 21 2015

A173410 The polynomial array L_q(n,k), evaluated at q = -1, i.e., L_{-1}(n,k).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 4, 4, 4, 1, 4, 4, 8, 4, 1, 8, 8, 20, 12, 6, 1, 8, 8, 28, 20, 18, 6, 1, 16, 16, 64, 48, 56, 24, 8, 1
Offset: 1

Views

Author

Mark Shattuck, Feb 17 2010

Keywords

Comments

L_q(n,k) is a q-generalization of the Lah number L(n,k) (see A105278) and is given by the two-term recurrence L_q(n,k) = L_q(n-1,k-1) + [(n-1)q + k_q]*L_q(n-1,k), for all positive integers n and k, with the boundary values L_q(n,0) = delta{n,0} and L_q(0,k) = delta_{k,0} for all nonnegative integers n and k, where m_q:=1+q+...+q^{m-1}.
Sequence above is L_q(n,k) evaluated at q = -1 (the nonzero values).
L_q(n,k) also arises as a distribution polynomial for a certain inversion statistic defined on the set of Lah distributions enumerated by L(n,k).

References

  • J. Lindsay, T. Mansour, and M. Shattuck, Generalizing a Relation between Polynomial Bases, Tech. Report, 2010.

Links

Crossrefs

See also A105278 and A000045.

Formula

Recurrence: The L_{-1}(n,k) are generated by the two-term recurrence L_{-1}(n,k)= L_{-1}(n-1,k-1) + [(n-1){-1}+k{-1}]*L_{-1}(n-1,k), for all positive integers n and k, with the boundary conditions as above for L_q(n,k), and where m_{-1} = [m is odd] for m a nonnegative integer.
Generating Function: 1 + sum_{n>=1} sum_{1 <= k <= n} L_{-1}(n,k)*x^n*y^k = (1 - 2x^2 - x^2y^2 + xy + 2x^3y^2 + 2x^2y - x^3y^3)/(1 - 2x^2 -2x^2y^2 - 2x^4y^2 + x^4y^4)
Connection Constant Relation: The L_{-1}(n,k) are also uniquely determined by the polynomial relations x*(x+1_{-1})*(x+2_{-1})*...*(x+(n-1){-1})= sum{k=1}^n L_{-1}(n,k)*x*(x-1_{-1})*...*(x-(k-1){-1}), for positive integers n, where m{-1} = [m is odd].
Row Sum: sum_{k=0}^n L_{-1}(n,k) = f_r, where r = [3*n/2] and f_m denotes the Fibonacci sequence given by the recurrence f_m=f_{m-1}+f_{m-2} if m >= 2, with f_0=f_1=1 (see A000045).

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