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 A247498 #48 Feb 26 2025 05:54:12
%S A247498 1,0,1,-1,1,1,0,0,2,1,5,-2,3,3,1,0,0,2,8,4,1,-61,16,-3,18,15,5,1,0,0,
%T A247498 2,32,52,24,6,1,1385,-272,63,48,165,110,35,7,1,0,0,2,128,484,480,198,
%U A247498 48,8,1,-50521,7936,-1383,528,1395,2000,1085,322,63,9,1
%N A247498 Generalized Euler numbers: Square array read by descending antidiagonals, T(n, k) = k!*[x^k] exp(n*x)*sech(x), n>=0, k>=0.
%C A247498 This two-dimensional array of numbers can be seen as a generalization of the Euler secant and Euler tangent numbers (which are in their compressed and signless form A000364 resp. A000182 or interleaved in A000111). The cases n=0 and n=1 reduce to their expanded and signed forms A122045 and A155585. Moreover the columns are the values of the Swiss-Knife polynomials A153641 evaluated at the nonnegative integers.
%C A247498 Subsequences [3,3,1], [8,4,1], [15,5,1], [24,6,1], [35,7,1], [48,8,1], [63,9,1] found in rows of this entry, as a triangular array, are present in the antidiagonals of Table 5 of the East and Gray reference (A244490), and some subsequences in the rows of Table 5 are found in the antidiagonals of this entry, including [3,2,1] and [1,1]. Equivalently, the first four columns of Table 5 are embedded in this entry viewed as a square array on the table page. An explicit formula with combinatorial interpretations for these numbers is provided in the reference, and others are known for the corresponding columns for the modified Hermite polynomials of A244490. - _Tom Copeland_, Oct 04 2016
%H A247498 J. East and R. D. Gray, <a href="http://arxiv.org/abs/1404.2359">Idempotent generators in finite partition monoids and related semigroups</a>, arXiv preprint arXiv:1404.2359 [math.GR], 2014.
%F A247498 G.f. for column k: the k-th column consists of the values of the k-th Swiss-Knife polynomial skp_{k}(x) evaluated at x = 0,1,2,...
%F A247498 O.g.f. for column k: Sum_{j=0..k} (-1)^(k+1)*A247501(k,j)/(x-1)^(j+1).
%e A247498 Square array starts:
%e A247498 [n\k][0][1] [2] [3] [4] [5] [6] [7] [8]
%e A247498 [0] 1, 0, -1, 0, 5, 0, -61, 0, 1385, ... A122045
%e A247498 [1] 1, 1, 0, -2, 0, 16, 0, -272, 0, ... A155585
%e A247498 [2] 1, 2, 3, 2, -3, 2, 63, 2, -1383, ... A119880
%e A247498 [3] 1, 3, 8, 18, 32, 48, 128, 528, 512, ... A119881
%e A247498 [4] 1, 4, 15, 52, 165, 484, 1395, 4372, 14505, ...
%e A247498 [5] 1, 5, 24, 110, 480, 2000, 8064, 32240, 130560, ... A225116
%e A247498 [6] 1, 6, 35, 198, 1085, 5766, 29855, 151878, 766745, ...
%e A247498 A000012, A001477, A067998, A121670, ...
%e A247498 Triangular array starts:
%e A247498 1,
%e A247498 0, 1,
%e A247498 -1, 1, 1,
%e A247498 0, 0, 2, 1,
%e A247498 5, -2, 3, 3, 1,
%e A247498 0, 0, 2, 8, 4, 1,
%e A247498 -61, 16, -3, 18, 15, 5, 1.
%p A247498 # EGF (row)
%p A247498 egf := n -> exp(n*x)*sech(x):
%p A247498 seq(print(seq(k!*coeff(series(egf(n),x,k+2),x,k),k=0..8)), n=0..6);
%p A247498 # Swiss-Knife polynomial (column)
%p A247498 SKP := proc(n, x) local v, k, A; A := k -> `if`(k mod 4 = 0,0,(-1)^iquo(k,4)); add(2^iquo(-k,2)*A(k+1)*add((-1)^v* binomial(k,v)*(v+x+1)^n,v=0..k), k=0..n); expand(%) end:
%p A247498 seq(print(seq(SKP(k, n), n=0..9)), k=0..6);
%p A247498 # OGF (column)
%p A247498 col := proc(n, len) local T; T := A247501_row(n);
%p A247498 (-1)^(n+1)*add(T[k+1]/(x-1)^(k+1),k=0..n);
%p A247498 seq(coeff(series(%,x,len+1),x,j),j=0..len) end:
%p A247498 seq(print(col(n,8)), n=0..6);
%t A247498 nmax = 10; Clear[row]; row[n_] := row[n] = CoefficientList[Exp[n*x]*Sech[x] + O[x]^(nmax+2), x][[1 ;; nmax+1]]*Range[0, nmax]!;
%t A247498 rows = Table[row[n], {n, 0, nmax}];
%t A247498 T[n_, k_] := rows[[n+1, k+1]];
%t A247498 Table[T[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Dec 03 2017 *)
%Y A247498 Cf. A122045, A155585, A119880, A119881, A225116, A000364, A000182, A000111, A067998, A121670, A153641, A247501.
%Y A247498 Cf. A244490.
%K A247498 tabl,sign
%O A247498 0,9
%A A247498 _Peter Luschny_, Dec 14 2014