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 A369435 #24 Apr 26 2024 18:59:57
%S A369435 1,1,1,1,2,1,1,6,3,1,1,26,15,4,1,1,150,111,28,5,1,1,1082,1095,292,45,
%T A369435 6,1,1,9366,13503,4060,605,66,7,1,1,94586,199815,70564,10845,1086,91,
%U A369435 8,1,1,1091670,3449631,1471708,243005,23826,1771,120,9,1,1,14174522,68062695,35810212,6534045,653406,45955,2696,153,10,1
%N A369435 Square array A(n, k) = n! * [t^n] (exp(t)/(1+k-k*exp(t))) for n >= 0 and k >= 0, read by antidiagonals upwards.
%C A369435 The following formulae are conjectures:
%C A369435 (1) det(A(0..n, k..k+n)) = (Product_{i=1..n} i!)^2 for k >= 0 and n >= 0.
%C A369435 (2) A(n, k) = 1 + k * (Sum_{i=0..n-1} binomial(n, i) * A(i, k)) for k >= 0 and
%C A369435 n > 0 with initial values A(0, k) = 1 for k >= 0.
%C A369435 (3) A(n, k) = (k+1)^n + k * (Sum_{i=0..n-2} binomial(n, i) * A(i, k) *
%C A369435 ((k+1)^(n-i) - (k+1) * k^(n-1-i))) for k >= 0 and n > 1 with initial values
%C A369435 A(n, k) = (k+1)^n for k >= 0 and n < 2.
%C A369435 (4) Let B(n, k) = (k!) * (Sum_{i=k..n} (i!) * S2(i, k) * S2(n+1, i+1)) for 0 <=
%C A369435 k <= n where S2(i, j) = A048993(i, j). Then holds:
%C A369435 (a) B(n, k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * A(n, i) for 0 <= k
%C A369435 <= n;
%C A369435 (b) E.g.f. of row n >= 0: exp(x) * (Sum_{k=0..n} B(n, k) * x^k / (k!)).
%F A369435 A(n, k) = Sum_{i=0..n} A163626(n, i) * (-k)^i for n >= 0 and k >= 0.
%F A369435 A(n, k) = Sum_{i=0..n} A028246(n+1, i+1) * k^i for n >= 0 and k >= 0.
%F A369435 E.g.f. of column k >= 0: exp(t) / (1 + k - k * exp(t)).
%F A369435 A(n, n) = Sum_{i=0..n} A163626(n, i) * (-n)^i = Sum_{i=0..n} A028246(n+1, i+1) * n^i for n >= 0.
%F A369435 Conjecture: A(n, n) = (n + 1) * A321189(n) for n >= 0. [This is true. - _Peter Luschny_, Apr 26 2024]
%F A369435 A(n, n) = A372312(n). - _Peter Luschny_, Apr 26 2024
%e A369435 Array A(n, k) starts:
%e A369435 n\k : 0 1 2 3 4 5 6 7 8
%e A369435 ================================================================================
%e A369435 0 : 1 1 1 1 1 1 1 1 1
%e A369435 1 : 1 2 3 4 5 6 7 8 9
%e A369435 2 : 1 6 15 28 45 66 91 120 153
%e A369435 3 : 1 26 111 292 605 1086 1771 2696 3897
%e A369435 4 : 1 150 1095 4060 10845 23826 45955 80760 132345
%e A369435 5 : 1 1082 13503 70564 243005 653406 1490587 3024008 5618169
%e A369435 6 : 1 9366 199815 1471708 6534045 21502866 58018051 135878520 286195833
%e A369435 7 : 1 94586 3449631 35810212
%e A369435 8 : 1 1091670 68062695
%e A369435 9 : 1 14174522
%e A369435 .
%e A369435 Triangle T(n, k) starts:
%e A369435 [0] 1;
%e A369435 [1] 1, 1;
%e A369435 [2] 1, 2, 1;
%e A369435 [3] 1, 6, 3, 1;
%e A369435 [4] 1, 26, 15, 4, 1;
%e A369435 [5] 1, 150, 111, 28, 5, 1;
%e A369435 [6] 1, 1082, 1095, 292, 45, 6, 1;
%e A369435 [7] 1, 9366, 13503, 4060, 605, 66, 7, 1;
%e A369435 [8] 1, 94586, 199815, 70564, 10845, 1086, 91, 8, 1;
%e A369435 [9] 1, 1091670, 3449631, 1471708, 243005, 23826, 1771, 120, 9, 1;
%p A369435 egf := exp(t) / (1 + x*(1 - exp(t))): sert := series(egf, t, 12):
%p A369435 col := k -> local j; seq(subs(x=k, j!*coeff(sert, t, j)), j=0..9):
%p A369435 T := (n, k) -> col(k)[n - k + 1]: # Triangle
%p A369435 for n from 0 to 9 do seq(T(n, k), k=0..n) od; # _Peter Luschny_, Jan 24 2024
%p A369435 with(combinat): # WP Worpitzky polynomials, WC coefficients of WP.
%p A369435 WC := (n, k) -> local j; add(eulerian1(n, j)*binomial(n-j, n-k), j=0..n):
%p A369435 WP := n -> local j; add(WC(n, j) * x^j, j=0..n):
%p A369435 A369435row := (n, k) -> subs(x = k, WP(n)):
%p A369435 seq(lprint(seq(A369435row(n, k), k = 0..7)), n = 0..7);
%p A369435 # _Peter Luschny_, Apr 26 2024
%o A369435 (PARI) {A(n, k) = n! * polcoeff(exp(x+x*O(x^n)) / (1+k-k*exp(x+x*O(x^n))), n)}
%Y A369435 Cf. A000012 (col 0 and row 0), A000629 (col 1), A201339 (col 2), A201354 (col 3), A201365 (col 4), A000027 (row 1), A000384 (row 2), A163626, A028246.
%Y A369435 Cf. A372312.
%K A369435 nonn,easy,tabl
%O A369435 0,5
%A A369435 _Werner Schulte_, Jan 23 2024