cp's OEIS Frontend

A355540 Triangle read by rows. Row n gives the coefficients of Product_{k=0..n} (x - k!) expanded in decreasing powers of x, with row 0 = {1}.

%I A355540 #49 Jul 10 2022 16:12:56
%S A355540 1,1,-1,1,-2,1,1,-4,5,-2,1,-10,29,-32,12,1,-34,269,-728,780,-288,1,
%T A355540 -154,4349,-33008,88140,-93888,34560,1,-874,115229,-3164288,23853900,
%U A355540 -63554688,67633920,-24883200,1,-5914,4520189,-583918448,15971865420,-120287210688,320383261440,-340899840000,125411328000
%N A355540 Triangle read by rows. Row n gives the coefficients of Product_{k=0..n} (x - k!) expanded in decreasing powers of x, with row 0 = {1}.
%C A355540 Essentially the same as A136457 with rows in reversed order.
%C A355540 Let M be an n X n matrix filled by Bell numbers A000110(j+k-2) with rows and columns j = 1..n, k = 1..n; then its determinant equals unsigned T(n, n). If we use A000110(j+k), the determinant will equal unsigned T(n+1, n). Can we find a general formula for T(n+m, n) based on determinants of matrices and Bell numbers?
%F A355540 T(n, 0) = 1.
%F A355540 T(n, 1) = -A003422(n).
%F A355540 T(n, 2) = Sum_{m=0..n-1} !m*m!.
%F A355540 T(n, k) = Sum_{m=0..n-1} -T(m, k-1)*m!.
%F A355540 T(n, n) = (-1)^n*A000178(n).
%F A355540 T(n, n-1) = -(-1)^n*A203227(n), for n > 0.
%F A355540 T(n+1, n) = (-1)^n*A000178(n)*A000522(n).
%F A355540 Sum_{m=0..k} T(n, k) = 0, for n > 0.
%F A355540 Sum_{m=0..k} abs(T(n, k)) = A217757(n+1).
%e A355540 The triangle begins:
%e A355540   1;
%e A355540   1,   -1;
%e A355540   1,   -2,      1;
%e A355540   1,   -4,      5,       -2;
%e A355540   1,  -10,     29,      -32,       12;
%e A355540   1,  -34,    269,     -728,      780,      -288;
%e A355540   1, -154,   4349,   -33008,    88140,    -93888,    34560;
%e A355540   1, -874, 115229, -3164288, 23853900, -63554688, 67633920, -24883200;
%e A355540   ...
%e A355540 Row 4: x^4 - 10*x^3 + 29*x^2 - 32*x + 12 = (x-0!)*(x-1!)*(x-2!)*(x-3!).
%e A355540 Illustration of T(1 to 5,1) as tree structure:
%e A355540 .
%e A355540 . o        o         o            o                         o
%e A355540 .          o         o            o                         o
%e A355540 .                   o o          o o                       o o
%e A355540 .                              ooo ooo                   ooo ooo
%e A355540 .                                             oooo oooo oooo oooo oooo oooo
%e A355540 . 1 +1 =   2 +2 =    4 +2*3 =     10 +6*4 =                 34
%e A355540 .
%e A355540 Illustration of T(2 to 4,2) as tree structure:
%e A355540 .
%e A355540 . o         o              -----o-----
%e A355540 .        o     o          o           o
%e A355540 .        o     o       ---o---     ---o---
%e A355540 .                     o   o   o   o   o   o
%e A355540 .                     o   o   o   o   o   o
%e A355540 .                    o o o o o o o o o o o o
%e A355540 . 1 +2*2 =  5 +6*4 =            29
%e A355540 .
%e A355540 Illustration of T(3 to 4,3) as tree structure:
%e A355540 .            ------------
%e A355540 . oo     ---o---      ---o---
%e A355540 .       o   o   o    o   o   o
%e A355540 .      o o o o o o  o o o o o o
%e A355540 .      o o o o o o  o o o o o o
%e A355540 .  2  +6*5 =      32
%o A355540 (PARI) T(n, k) = polcoeff(prod(m=0, n-1, (x-m!)), n-k);
%Y A355540 Cf. A000110, A000178, A000522, A003422, A136457, A203227, A217757.
%Y A355540 Cf. A008276 (The Stirling numbers of the first kind in reverse order).
%Y A355540 Cf. A039758 (Coefficients for polynomials with roots in odd numbers).
%Y A355540 Cf. A349226 (Coefficients for polynomials with roots in x^x).
%K A355540 sign,tabl
%O A355540 0,5
%A A355540 _Thomas Scheuerle_, Jul 06 2022