OFFSET
1,2
COMMENTS
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
LINKS
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.
FORMULA
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
EXAMPLE
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
MAPLE
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
MATHEMATICA
Flatten[ Table[ Sum[(-1)^{r + j}(2r + 1)(r^2 + r)^n/((r + j + 1)!(j - r)!), {r, j}], {n, 10}, {j, n}]]
PROG
(PARI) {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 */
(Magma) [[(&+[(-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
(Sage) [[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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Jun 16 2002
STATUS
approved