A080248 -id:A080248 - 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

A124373 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k(k+1)/2x).

Original entry on oeis.org

1, 1, 2, 6, 25, 135, 909, 7417, 71698, 806968, 10427825, 152915697, 2519879761, 46276398129, 940296067422, 21007099850230, 513172107841525, 13640345170943527, 392780078386164389, 12204609567437300313, 407757149671568266678, 14600807659376773500696

Offset: 0

Paul D. Hanna, Oct 28 2006

nonn

Starting (1, 2, 6, 25, ...) = row sums of triangle A080248. - Gary W. Adamson, Jul 11 2011

			Also generated by iterated binomial transforms in the following way:
  [1,2,6,25,135,909,7417,71698,...]  = binomial([1,1,3,12,64,433,3567,...]);
  [1,3,12,64,433,3567,34905,...]     = binomial^2([1,1,4,20,129,1045,...]);
  [1,4,20,129,1045,10209,117069,...] = binomial^3([1,1,5,30,226,2121,...]);
  [1,5,30,226,2121,23919,314605,...] = binomial^4([1,1,6,42,361,3835,...]);
  [1,6,42,361,3835,48885,724569,...] = binomial^5([1,1,7,56,540,6385,...]);
  [1,7,56,540,6385,90519,1490457,..] = binomial^6([1,1,8,72,769,9993,...]);
etc.

Alois P. Heinz, Table of n, a(n) for n = 0..358

Cf. A000110, A000217, A080248.

b:= proc(n, m) option remember; `if`(n=0, 1,
      b(n-1, m)*m*(m+1)/2 +b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 10 2019

b[n_, m_] := b[n, m] = If[n == 0, 1, b[n-1, m] m(m+1)/2 + b[n-1, m+1]];
a[n_] := b[n, 0];
a /@ Range[0, 23] (* Jean-François Alcover, Nov 02 2020, after Alois P. Heinz *)

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

O.g.f.: A(x) = 1 + x/(1-x) + x^2/((1-x)*(1-3x)) + x^3/((1-x)*(1-3x)*(1-6x)) + x^3/((1-x)*(1-3x)*(1-6x)*(1-10x)) + ...

G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-(k+1)*(k+2)*x/2)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2013

A080249 Stirling-like number triangle defined by the sequence A000292=C(n+3,3).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 21, 15, 1, 1, 85, 171, 35, 1, 1, 341, 1795, 871, 70, 1, 1, 1365, 18291, 19215, 3321, 126, 1, 1, 5461, 184275, 402591, 135450, 10377, 210, 1, 1, 21845, 1848211, 8236095, 5143341, 716562, 28017, 330, 1, 1, 87381, 18503955, 166570111, 188253030, 45270813, 3069990, 67617, 495, 1

Offset: 0

Paul Barry, Feb 17 2003

Columns include A002450, A016225. The defining sequence A000292=C(n+3,3) is the sequence of partial sums of the defining sequence for number triangle A080248.

			Triangle begins:
1;
1,    1;
1,    5,      1;
1,   21,     15,      1;
1,   85,    171,     35,      1;
1,  341,   1795,    871,     70,     1;
1, 1365,  18291,  19215,   3321,   126,   1;
1, 5461, 184275, 402591, 135450, 10377, 210, 1;
For example, 171 = 21+10*15, 35 = 15+20*1.

Cf. A080248, A008277.

T(n,k) = T(n-1,k-1) + A000292(k)*T(n-1,k). Columns are generated by 1/product{k=0..n, 1-C(k+3,3)*x}.

A226941 Expansion of 1/((1-x)(1-3x)(1-6x)(1-10x)(1-15x)).

Original entry on oeis.org

1, 35, 798, 15178, 262739, 4310073, 68451856, 1065454016, 16372593237, 249520885471, 3782278181474, 57129692163414, 860905800344695, 12953222527379429, 194694881199600852, 2924389779305546572, 43905519073297744313, 658979550560400579147, 9888661146758667705190

Offset: 1

Yahia Kahloune, Jun 23 2013

nonn

Note that the denominator has 5 triangular numbers: 1, 3, 6, 10, and 15.

Index entries for linear recurrences with constant coefficients, signature (35,-427,2193,-4500,2700).

Column k = 4 of A080248. Cf. A003462, A016211, A021514.

a(n) = (15^(n+4) - 6*10^(n+4) + 14*6^(n+4) - 15*3^(n+4) + 6)/7560.

a(n) = 35*a(n-1) - 427*a(n-2) + 2193*a(n-3) - 4500*a(n-4) + 2700*a(n-5) for n > 5. - Chai Wah Wu, Aug 10 2020

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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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A124373 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k*(k+1)/2*x).

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A080249 Stirling-like number triangle defined by the sequence A000292=C(n+3,3).

Views

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Examples

Crossrefs

Formula

A226941 Expansion of 1/((1-x)(1-3x)(1-6x)(1-10x)(1-15x)).

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Author

Keywords

Comments

Links

Crossrefs

Formula

A124373 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k(k+1)/2x).