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.
%I A285066 #31 Mar 10 2025 11:12:09
%S A285066 1,1,4,1,24,32,1,124,480,384,1,624,5312,10752,6144,1,3124,52800,
%T A285066 203520,276480,122880,1,15624,500192,3279360,7956480,8110080,2949120,
%U A285066 1,78124,4626720,48633984,187729920,329441280,268369920,82575360,1,390624,42265472,687762432,3969552384,10672865280,14615838720,9909043200,2642411520,1,1953124,383514240,9448097280,78486589440,303521218560,621544734720,696605736960,404288962560,95126814720
%N A285066 Triangle read by rows: T(n, m) = A285061(n, m)*m!, 0 <= m <= n.
%C A285066 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.
%C A285066 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}.
%C A285066 The diagonal sequence is A047053. The row sums give A285067. The alternating sum of row n is A141413(n+2), n >= 0.
%C A285066 The first column sequences are: A000012, 4*A003463, 2!*4^2*A016234.
%H A285066 Wolfdieter Lang, <a href="https://arxiv.org/abs/1707.04451">On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers</a>, arXiv:1707.04451 [math.NT], 2017.
%F A285066 T(n, m) = A285061(n, m)*m! = A111578(n, m)*(4^m*m!), 0 <= m <= n.
%F A285066 T(n, m) = Sum_{k=0..n} binomial(m,k)*(-1)^(k-m)*(1+4*k)^n.
%F A285066 T(n, m) = Sum_{j=0..n} binomial(n-j,m-j)*A225118(n,n-j).
%F A285066 Recurrence: T(n, -1) = 0, T(0, 0) = 1, T(n, m) = 0 if n < m and T(n, m) =
%F A285066 4*m*T(n-1, m-1) + (1+4*m)*T(n-1, m) for n >= 1, m=0..n.
%F A285066 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)).
%F A285066 E.g.f. column m: exp(x)*(exp(4*x) - 1)^m, m >= 0.
%F A285066 O.g.f. column m: m!*(4*x)^m/Product_{j=0..m} (1 - (1 + 4*j)*x), m >= 0.
%e A285066 The triangle T(n, m) begins:
%e A285066 n\m 0 1 2 3 4 5 6 7
%e A285066 0: 1
%e A285066 1: 1 4
%e A285066 2: 1 24 32
%e A285066 3: 1 124 480 384
%e A285066 4: 1 624 5312 10752 6144
%e A285066 5: 1 3124 52800 203520 276480 122880
%e A285066 6: 1 15624 500192 3279360 7956480 8110080 2949120
%e A285066 7: 1 78124 4626720 48633984 187729920 329441280 268369920 82575360
%e A285066 ...
%e A285066 row 8: 1 390624 42265472 687762432 3969552384 10672865280 14615838720 9909043200 2642411520
%e A285066 row 9: 1 1953124 383514240 9448097280 78486589440 303521218560 621544734720 696605736960 404288962560 95126814720
%e A285066 ...
%t A285066 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 *)
%o A285066 (Python)
%o A285066 from sympy import binomial
%o A285066 def T(n, m):
%o A285066 return sum([binomial(m, k)*(-1)**(k - m)*(1 + 4*k)**n for k in range(n + 1)])
%o A285066 for n in range(21):
%o A285066 print([T(n, m) for m in range(n + 1)])
%o A285066 # _Indranil Ghosh_, May 02 2017
%Y A285066 Cf. A000012, 4*A003463, 32*A016234, A111578 (T(n, m)/4^m), A131689, A141413, A145901, A225118, A225472, A225473, A284861, A285061.
%K A285066 nonn,tabl,easy
%O A285066 0,3
%A A285066 _Wolfdieter Lang_, Apr 19 2017