0,7

T(n, n)= 1; T(n+1, n)= n; T(n+2, n)= A002061(n+1)= n^2 + n + 1; T(n+3, n)= n^3 + 3*n^2 + 5*n + 2.

T(n, k) = (k + 1)*T(n, k + 1)-T(n-1, k); T(n, n)= 1; T(n, k)= 0, if k>n. T(n, k) = (n-1)*T(n-1, k) + (n-k-1)*T(n-2, k). k!*T(n, k) = A068106(n+1, k+1). Sum(k>=0; T(n, k) = A003470(n+1).

T(n, k) = 1/k! * Sum_{j>=0} (-1)^j*binomial(n-k, j)*(n-j)!. - Philippe DELEHAM, Jun 13 2005

Comments from Peter Bala (pbala(AT)toucansurf.com), Aug 14 2008 (Start): The following remarks all relate to the array read as a square array: e.g.f for column k: exp(-y)/(1-y)^(k+1); e.g.f. for array: exp(-y)/(1-x-y) = (1 + x + x^2 + x^3 + ...) + (x + 2*x^2 + 3*x^3 + 4*x^4 + ...)*y + (1 + 3*x + 7*x^2 + 13*x^3 + ...)*y^2/2! + ... .

This table is closely connected to the constant e. The row, column and diagonal entries of this table occur in series formulas for e.

Row n for n >= 2: e = n!*(1/T(n,0) + (-1)^n*[1/(1!*T(n,0)*T(n,1)) + 1/(2!*T(n,1)*T(n,2)) + 1/(3!*T(n,2)*T(n,3)) + ...]). For example, row 3 gives e = 6*(1/2 - 1/(1!*2*11) - 1/(2!*11*32) - 1/(3!*32*71) - ...). See A095000.

Column 0: e = 2 + sum {n = 2..inf} (-1)^n*n!/(T(n,0)*T(n+1,0)) = 2 + 2!/(1*2) - 3 !/(2*9) + 4!/(9*44) - ... .

Column k, k >= 1: e = (1+1/1!+1/2!+...+1/k!)+ 1/k!*sum {n = 0..inf} (-1)^n*n!/(T(n,k)*T(n+1,k)). For example, column 3 gives e = 8/3 + 1/6*[1/(1*3) - 1/(3*13) + 2/(13*71) - 6/(71*465) + ...].

Main diagonal: e = 1 + 2*[1/(1*1) - 1/(1*7) + 1/(7*71) - 1/(71*1001) + ...].

First subdiagonal: e = 8/3 + 5/(3*32) - 7/(32*465) + 9/(465*8544) - ... .

Second subdiagonal: e = 2*[1 + 2^2/(1*11) - 3^2/(11*181) + 4^2/(181*3539) - ...]. See A143413.

Third subdiagonal: e = 3 - (2*3*5)/(2*53) + (3*4*7)/(53*1214) - (4*5*9)/(1214*30637) + ... .

For the corresponding results for the constants 1/e, sqrt(e) and 1/sqrt(e) see A143409, A143410 and A143411 respectively. For other arrays similarly related to constants see A008288 (for log(2)), A108625 (for zeta(2)) and A143007 (for zeta(3)). (End)

1; 0, 1; 1, 1, 1; 2, 3, 2, 1; 9, 11, 7, 3, 1; 44, 53, 32, 13, 4, 1; ...

Formatted as a square array:

1 3 7 13 21 31 43 57 which equals A002061

2 11 32 71 134 227 356 which equals A094792

9 53 181 465 1001 1909 which equals A094793

44 309 1214 3539 8544 which equals A094794

265 2119 9403 30637 which equals A023043

1854 16687 82508 which equals A023044

14833 148329 which equals A023045

Formatted as a triangular array (mirror of A076731):

1

0 1

1 1 1

2 3 2 1

9 11 7 3 1

44 53 32 13 4 1

265 309 181 71 21 5 1

1854 2119 1214 465 134 31 6 1

14833 16687 9403 3539 1001 227 43 7 1

133496 148329 82508 30637 8544 1909 356 57 8 1

Columns: A000166, A000155, AOOO153, A000261, A001909, A001910.

Cf. A068106 A003470 AOO2061. Mirror image of A076731.

Cf. A143409, A143410, A143411, A143413.

Sequence in context: A062323 A020858 A090664 this_sequence A076224 A114729 A144365

Adjacent sequences: A086761 A086762 A086763 this_sequence A086765 A086766 A086767

easy,nonn,tabl

DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Aug 02 2003

More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Mar 28 2005

Additional comments from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 30 2006