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

0,5

Comments

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?

Links

Table of n, a(n) for n=0..44.

Formula

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).

Example

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

Prog

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

Crossrefs

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).

Sequence in context: A198895 A355635 A118686 * A102610 A203300 A134172

Adjacent sequences: A355537 A355538 A355539 * A355541 A355542 A355543

Keyword

sign,tabl

Author

Thomas Scheuerle, Jul 06 2022

Status

approved