A135921 -id:A135921 - cp's OEIS frontend

A071951 Triangle of Legendre-Stirling numbers of the second kind T(n,j), n >= 1, 1 <= j <= n, read by rows.

1, 2, 1, 4, 8, 1, 8, 52, 20, 1, 16, 320, 292, 40, 1, 32, 1936, 3824, 1092, 70, 1, 64, 11648, 47824, 25664, 3192, 112, 1, 128, 69952, 585536, 561104, 121424, 7896, 168, 1, 256, 419840, 7096384, 11807616, 4203824, 453056, 17304, 240, 1, 512, 2519296, 85576448, 243248704, 137922336, 23232176, 1422080, 34584, 330, 1

Offset: 1

N. J. A. Sloane, Jun 16 2002

Removing a factor of 2^m from the m-th subdiagonal (the main diagonal corresponds to m = 0) gives the triangle A080248. - Peter Bala, Oct 15 2023

			The triangle begins:
n\j   1      2       3        4       5      6     7   8 9 ...
1:    1
2:    2      1
3:    4      8       1
4:    8     52      20        1
5:   16    320     292       40       1
6:   32   1936    3824     1092      70      1
7:   64  11648   47824    25664    3192    112     1
8:  128  69952  585536   561104  121424   7896   168   1
9:  256 419840 7096384 11807616 4203824 453056 17304 240 1
...
Row 10: 512 2519296 85576448 243248704 137922336 23232176 1422080 34584 330 1. Reformatted by _Wolfdieter Lang_, Apr 10 2013

Robert Israel, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)
G. E. Andrews, W. Gawronski and L. L. Littlejohn, The Legendre-Stirling Numbers, Discrete Mathematics, Volume 311, Issue 14, 28 July 2011, Pages 1255-1272.
M. W. Coffey, M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.
R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, arXiv:1302.4694 [math.CO], 2013.
R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, Europ. J. Combin., 43, 2015, 55-67.
José L. Cereceda, A refinement of Lang's formula for the sum of powers of integers, arXiv:2301.02141 [math.NT], 2023.
E. S. Egge, Legendre-Stirling permutations, Eur. J. Combin. 13 (2010) 1735-1750.
W. N. Everitt, L. L. Littlejohn and R. Wellman, Legendre polynomials, Legendre-Stirling numbers and the left-definite spectral analysis of the Legendre differential expression, J. Comput. Appl. Math. 148, 2002, 213-238.
H. Li, T. MacHenry, Permanents and Determinants, Weighted Isobaric Polynomials, and Integer Sequences, J. Int. Seq. 16 (2013) #13.3.5, Theorem 43.
L. L. Littlejohn and R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181(2), 2002, 280-339.

Diagonals give A007290, A000079, A016129, A016309.
The column sequences are A000079 (powers of 2), A016129, A016309, A071952, A089274, A089277.
Cf. A080248, A089278, A089500, A160562.

[[(&+[(-1)^(r+j)*(2*r+1)*(r^2+r)^n/(Factorial(r+j+1)*Factorial(j-r)): r in [1..j]]): j in [1..n]]: n in [1..12]]; // G. C. Greubel, Mar 16 2019

N:= 20: # to get the first N rows, flattened
for j from 1 to N do S[j]:= series(x^j/mul(1-r*(r+1)*x, r=1..j), x, N+1) od:
seq(seq(coeff(S[j],x,i),j=1..i),i=1..N); # Robert Israel, Dec 03 2015
# alternative
A071951 := proc(n,k)
    option remember;
    if k =0 then
        if n = 0 then
            1;
        else
            0;
        end if;
    elif n = 0 then
        if k =0 then
            1;
        else
            0;
        end if;
    else
        procname(n-1,k-1)+k*(k+1)*procname(n-1,k) ;
    end if;
end proc: # R. J. Mathar, Jun 30 2018

Flatten[ Table[ Sum[(-1)^{r + j}(2r + 1)(r^2 + r)^n/((r + j + 1)!(j - r)!), {r, j}], {n, 10}, {j, n}]]

{T(n, k) = sum( i=0, k, (-1)^(i+k) * (2*i + 1) * (i*i + i)^n / (k-i)! / (k+i+1)! )} /* Michael Somos, Feb 25 2012 */

[[sum( (-1)^(r+j)*(2*r+1)*(r^2+r)^n/(factorial(r+j+1)*factorial(j-r)) for r in (1..j)) for j in (1..n)] for n in (1..12)] # G. C. Greubel, Mar 16 2019

T(n, j) = Sum_{r=1..j} (-1)^(r+j)*(2*r+1)*(r^2+r)^n/((r+j+1)!*(j-r)!).
G.f. for j-th column (without leading zeros): 1/Product_{r=1..j} (1 - r*(r+1)*x), j >= 1. From eq.(4.5) of the Everitt et al. paper.
A135921(n+1) = row sums. - Michael Somos, Feb 25 2012
Sum_{n=j..m} binomial(m,n)*T(n,j)*4^(n-j) = A160562(m,j) for 1 <= j <= m. - Werner Schulte, Dec 03 2015

A135920 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k^2*x).

Original entry on oeis.org

1, 1, 2, 7, 37, 264, 2433, 27913, 386906, 6346119, 121159373, 2655174768, 66028903633, 1845579100993, 57506847262162, 1983312152411351, 75238783332550789, 3122408658986242072, 141063757638078429489

Offset: 0

Paul D. Hanna, Dec 06 2007

nonn

From Peter Bala, Sep 27 2012: (Start)
Generalized Bell numbers; row sums of A036969.
a(n) is equal to the number of partitions of the set {1,1',2,2',...,n,n'} into disjoint nonempty subsets V1,...,Vk (1 <= k <= n) such that, for each 1 <= j <= k, if i is the least integer such that either i or i' belongs to Vj then {i,i'} is a subset of Vj.
Example: a(3) = 7: There is a single partition into one set {1,1',2,2',3,3'}; five partitions into two sets, namely, {1,1',2,2'}{3,3'}, {1,1',3,3'}{2,2'}, {1,1'}{2,2',3,3'}, {1,1',3}{2,2',3'} and {1,1',3'}{2,2',3}; and finally a single partition into three sets {1,1'}{2,2'}{3,3'}. (End)

			O.g.f.: A(x) = 1 + x/(1-x) + x^2/((1-x)*(1-4x)) + x^3/((1-x)*(1-4x)*(1-9x)) + x^4/((1-x)*(1-4x)*(1-9x)*(1-16x)) + ...
Also generated by iterated binomial transforms in the following way:
[1,2,7,37,264,2433,27913,...] = BINOMIAL([1,1,4,21,151,1422,16629,..]);
[1,4,21,151,1422,16629,234529,...] = BINOMIAL^3([1,1,6,43,393,4596,...]);
[1,6,43,393,4596,66049,1125905,...] = BINOMIAL^5([1,1,8,73,811,11274,...]);
[1,8,73,811,11274,191685,...] = BINOMIAL^7([1,1,10,111,1453,23328,...]);
[1,10,111,1453,23328,456033,...] = BINOMIAL^9([1,1,12,157,2367,43014,...]);
etc.

S. Matsumoto and J. Novak, Jucys-Murphy Elements and Unitary Matrix Integrals arXiv.0905.1992 [math.CO], 2009-2012.

Cf. A135921, A124373, A000110, A036969.

nmax = 20;
A[x_] = Sum[x^n/Product[1 - k^2 x, {k, 0, n}], {n, 0, nmax}];
CoefficientList[A[x] + O[x]^nmax, x] (* Jean-François Alcover, Jul 27 2018 *)

a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-j^2*x+x*O(x^n))), n)

From Peter Bala, Sep 27 2012: (Start)
Let E(x) = cosh(sqrt(2*x)) = Sum_{n >= 0} x^n/((2*n)!/2^n). A generating function is E((E(x) - 1)) = 1 + x + 2*x^2/6 + 7*x^3/90 + ..., where the sequence of denominators [1, 1, 6, 90, ...] is given by (2*n)!/2^n. Cf. A000110 which has generating function exp((exp(x) - 1)).
An e.g.f. is E((E(x^2/2) - 1)) = 1 + x^2/2! + 2*x^4/4! + 7*x^6/6! + .... (End)
G.f.: 1 + x/(U(0)-x) where U(k) = 1 - 2*x*k - x*k^2 + x*(x*(k+1)^2 - 1)/U(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Oct 11 2012
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-k^2*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
Conjecture: a(n) = A369527(n-1, 0) = A369595(n-1, 0) for n > 0 with a(0) = 1. - Mikhail Kurkov, Apr 25 2024

A191935 Triangle read by rows of Legendre-Stirling numbers of the second kind.

Original entry on oeis.org

1, 1, 2, 1, 8, 4, 1, 20, 52, 8, 1, 40, 292, 320, 16, 1, 70, 1092, 3824, 1936, 32, 1, 112, 3192, 25664, 47824, 11648, 64, 1, 168, 7896, 121424, 561104, 585536, 69952, 128, 1, 240, 17304, 453056, 4203824, 11807616, 7096384, 419840, 256, 1, 330, 34584, 1422080, 23232176, 137922336, 243248704, 85576448, 2519296, 512

Offset: 1

N. J. A. Sloane, Jun 19 2011

			Triangle begins:
  1;
  1   2;
  1   8    4;
  1  20   52      8;
  1  40  292    320     16;
  1  70 1092   3824   1936     32;
  1 112 3192  25664  47824  11648    64;
  1 168 7896 121424 561104 585536 69952 128;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
G. E. Andrews, W. Gawronski and L. L. Littlejohn, The Legendre-Stirling Numbers
G. E. Andrews et al., The Legendre-Stirling numbers, Discrete Math., 311 (2011), 1255-1272.

Cf. A135921 (row sums), A191936.

Mirror of triangle A071951. - Omar E. Pol, Jan 10 2012

Ps[n_, k_]:= Sum[(-1)^(j+k)*(2*j+1)*j^n*(1+j)^n/((j+k+1)!*(k-j)!), {j,0,k}];
Table[Ps[n, n-k+1], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jun 06 2021 *)

T071951(n, k) = sum(i=0, k, (-1)^(i+k) * (2*i + 1) * (i*i + i)^n / (k-i)! / (k+i+1)! );
for (n=1, 10, for (k=1, n, print1(T071951(n,n-k+1), ", ")); print); \\ Michel Marcus, Nov 24 2019

def Ps(n,k): return sum( (-1)^(j+k)*(2*j+1)*j^n*(1+j)^n/(factorial(j+k+1) * factorial(k-j)) for j in (0..k) )
flatten([[Ps(n,n-k+1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jun 06 2021

From G. C. Greubel, Jun 06 2021: (Start)
T(n, k) = Ps(n, n-k+1), where Ps(n, k) = Sum_{j=0..k} (-1)^(j+k)*(2*j+1)*j^n*(1 + j)^n/((j+k+1)!*(k-j)!).
Sum_{k=1..n} T(n, k) = A135921(n). (End)

More terms from Omar E. Pol, Jan 10 2012

More terms from Michel Marcus, Nov 24 2019

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A071951 Triangle of Legendre-Stirling numbers of the second kind T(n,j), n >= 1, 1 <= j <= n, read by rows.

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A135920 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k^2*x).

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A191935 Triangle read by rows of Legendre-Stirling numbers of the second kind.

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