OFFSET
0,3
COMMENTS
This is the Sheffer triangle S2[4,1] = A285061 with column m scaled by m!. This is the fourth member of the triangle family A131689, A145901 and A284861.
This triangle appears in the o.g.f. G(n, x) = Sum_{m=0..n} T(n, m)*x^m/(1-x)^(m+1), n >= 0, of the power sequence {(1+4*m)^n}_{m >= 0}.
LINKS
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.
FORMULA
T(n, m) = Sum_{k=0..n} binomial(m,k)*(-1)^(k-m)*(1+4*k)^n.
T(n, m) = Sum_{j=0..n} binomial(n-j,m-j)*A225118(n,n-j).
Recurrence: T(n, -1) = 0, T(0, 0) = 1, T(n, m) = 0 if n < m and T(n, m) =
4*m*T(n-1, m-1) + (1+4*m)*T(n-1, m) for n >= 1, m=0..n.
E.g.f. row polynomials R(n, x) = Sum_{m=0..n} T(n, m)*x^m: exp(z)/(1 - x*(exp(4*z) - 1).
E.g.f. column m: exp(x)*(exp(4*x) - 1)^m, m >= 0.
O.g.f. column m: m!*(4*x)^m/Product_{j=0..m} (1 - (1 + 4*j)*x), m >= 0.
EXAMPLE
The triangle T(n, m) begins:
n\m 0 1 2 3 4 5 6 7
0: 1
1: 1 4
2: 1 24 32
3: 1 124 480 384
4: 1 624 5312 10752 6144
5: 1 3124 52800 203520 276480 122880
6: 1 15624 500192 3279360 7956480 8110080 2949120
7: 1 78124 4626720 48633984 187729920 329441280 268369920 82575360
...
row 8: 1 390624 42265472 687762432 3969552384 10672865280 14615838720 9909043200 2642411520,
row 9: 1 1953124 383514240 9448097280 78486589440 303521218560 621544734720 696605736960 404288962560 95126814720,
row n=10: 1 9765624 3467253152 127427435520 1485441576960 7943677378560 22823127613440 37604818944000 35632919347200 18074094796800 3805072588800.
...
MATHEMATICA
T[n_, m_]:=Sum[Binomial[m, k]*(-1)^(k - m)*(1 + 4k)^n, {k, 0, n}]; Table[T[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Indranil Ghosh, May 02 2017 *)
PROG
(Python)
from sympy import binomial
def T(n, m):
return sum([binomial(m, k)*(-1)**(k - m)*(1 + 4*k)**n for k in range(n + 1)])
for n in range(21):
print([T(n, m) for m in range(n + 1)])
# Indranil Ghosh, May 02 2017
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Apr 19 2017
STATUS
approved