A136457 -id:A136457 - 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}.

1, 1, -1, 1, -2, 1, 1, -4, 5, -2, 1, -10, 29, -32, 12, 1, -34, 269, -728, 780, -288, 1, -154, 4349, -33008, 88140, -93888, 34560, 1, -874, 115229, -3164288, 23853900, -63554688, 67633920, -24883200, 1, -5914, 4520189, -583918448, 15971865420, -120287210688, 320383261440, -340899840000, 125411328000

Offset: 0

Thomas Scheuerle, Jul 06 2022

Essentially the same as A136457 with rows in reversed order.

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?

			The triangle begins:
  1;
  1,   -1;
  1,   -2,      1;
  1,   -4,      5,       -2;
  1,  -10,     29,      -32,       12;
  1,  -34,    269,     -728,      780,      -288;
  1, -154,   4349,   -33008,    88140,    -93888,    34560;
  1, -874, 115229, -3164288, 23853900, -63554688, 67633920, -24883200;
  ...
Row 4: x^4 - 10*x^3 + 29*x^2 - 32*x + 12 = (x-0!)*(x-1!)*(x-2!)*(x-3!).
Illustration of T(1 to 5,1) as tree structure:
.
. o        o         o            o                         o
.          o         o            o                         o
.                   o o          o o                       o o
.                              ooo ooo                   ooo ooo
.                                             oooo oooo oooo oooo oooo oooo
. 1 +1 =   2 +2 =    4 +2*3 =     10 +6*4 =                 34
.
Illustration of T(2 to 4,2) as tree structure:
.
. o         o              -----o-----
.        o     o          o           o
.        o     o       ---o---     ---o---
.                     o   o   o   o   o   o
.                     o   o   o   o   o   o
.                    o o o o o o o o o o o o
. 1 +2*2 =  5 +6*4 =            29
.
Illustration of T(3 to 4,3) as tree structure:
.            ------------
. oo     ---o---      ---o---
.       o   o   o    o   o   o
.      o o o o o o  o o o o o o
.      o o o o o o  o o o o o o
.  2  +6*5 =      32

Cf. A000110, A000178, A000522, A003422, A136457, A203227, A217757.
Cf. A008276 (The Stirling numbers of the first kind in reverse order).
Cf. A039758 (Coefficients for polynomials with roots in odd numbers).
Cf. A349226 (Coefficients for polynomials with roots in x^x).

T(n, k) = polcoeff(prod(m=0, n-1, (x-m!)), n-k);

T(n, 0) = 1.
T(n, 1) = -A003422(n).
T(n, 2) = Sum_{m=0..n-1} !m*m!.
T(n, k) = Sum_{m=0..n-1} -T(m, k-1)*m!.
T(n, n) = (-1)^n*A000178(n).
T(n, n-1) = -(-1)^n*A203227(n), for n > 0.
T(n+1, n) = (-1)^n*A000178(n)*A000522(n).
Sum_{m=0..k} T(n, k) = 0, for n > 0.
Sum_{m=0..k} abs(T(n, k)) = A217757(n+1).

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}.

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