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A028421 Triangle read by rows: f(n, k) = (k+1)*A132393(n+1, k+1), for n >= 0, k = 0, 1, ..., n. 31
1, 1, 2, 2, 6, 3, 6, 22, 18, 4, 24, 100, 105, 40, 5, 120, 548, 675, 340, 75, 6, 720, 3528, 4872, 2940, 875, 126, 7, 5040, 26136, 39396, 27076, 9800, 1932, 196, 8, 40320, 219168, 354372, 269136, 112245, 27216, 3822, 288, 9, 362880, 2053152 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Previous name was: Number triangle f(n, k) from n-th differences of the sequence {1/m^2}_{m >= 1}, for n >= 0; the n-th difference sequence is {(-1)^n*n!*P(n, m)/D(n, m)^2}_{m >= 1} where P(n, x) is the row polynomial P(n, x) = Sum_{k=0..n} f(n,k)*x^k and D(n, x) = x*(x+1)*...*(x+n).

From Johannes W. Meijer, Oct 07 2009: (Start)

The higher-order exponential integrals E(x,m,n) are defined in A163931 and the general formula of the asymptotic expansion of E(x,m,n) can be found in A163932.

We used the general formula and the asymptotic expansion of E(x,m=1,n), see A130534, to determine that E(x,m=2,n) ~ (exp(-x)/x^2)*(1 - (1+2*n)/x + (2 + 6*n + 3*n^2)/x^2 - (6 + 22*n + 18*n^2 + 4*n^3)/x^3 + ... ) which can be verified with the EA(x,2,n) formula, see A163932. The coefficients in the denominators of this expansion lead to the sequence given above.

The asymptotic expansion of E(x,m=2,n) leads for n from one to ten to known sequences, see the cross-references. With these sequences one can form the triangles A165674 (left hand columns) and A093905 (right hand columns).

(End)

For connections to an operator relation between log(x) and x^n(d/dx)^n, see A238363. - Tom Copeland, Feb 28 2014

From Wolfdieter Lang, Nov 25 2018: (Start)

The signed triangle t(n, k) := (-1)^{n-k}*f(n, k) gives (n+1)*N(-1;n,x) = Sum_{k=0..n} t(n, k)*x^k, where N(-1;n,x) are the Narumi polynomials with parameter a = -1 (see the Weisstein link).

The members of the n-th difference sequence of the sequence {1/m^2}_{m>=1} mentioned above satisfies the recurrence delta(n, m) = delta(n-1, m+1) - delta(n-1, m), for n >= 1, m >= 1, with input delta(0, m) = 1/m^2. The solution is delta(n, m) = (n+1)!*N(-1;n,-m)/risefac(m, n+1)^2, with Narumi polynomials N(-1;n,x) and the rising factorials risefac(x, n+1) = D(n, x) = x*(x+1)*...*(x+n).

The above mentioned row polynomials P satisfy P(n, x) = (-1)^n*(n + 1)*N(-1;n,-x), for n >= 0. The recurrence is P(n, x) = (-x^2*P(n-1, x+1) + (n+x)^2*P(n-1, x))/n, for n >= 1, and P(0, x) = 1. (End)

The triangle is the exponential Riordan square (cf. A321620) of -ln(1-x) with an additional main diagonal of zeros. - Peter Luschny, Jan 03 2019

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Eric Weisstein's World of Mathematics, Narumi polynomials [here for a = -1]

FORMULA

f(n, k) = (k+1)*s(n+1, k+1) (for n >= k) where s(n, k) is an unsigned Stirling number of the first kind. s(n, k) = A132393(n, k).

E.g.f.: d/dt(-log(1-t)/(1-t)^x). - Vladeta Jovovic, Oct 12 2003

The e.g.f. with offset 1:  y = x + (1 + 2*t)*x^2/2! + (2 + 6*t + 3*t^2)*x^3/3! + ... has series reversion with respect to x equal to y - (1 + 2*t)*y^2/2! + (1 + 3*t)^2*y^3/3! - (1 + 4*t)^3*y^4/4! + .... This is an e.g.f. for a signed version of A139526. - Peter Bala, Jul 18 2013

Recurrence: f(n, k) = 0 if n < k; if k = 0 then f(0, 0) = 1 and f(n, 0) = n* f(n-1, 0) for n >= 1, otherwise f(n, k) =  n*f(n-1, k) + ((k+1)/k)*f(n-1, k-1). From the unsigned Stirling1 recurrence. - Wolfdieter Lang, Nov 25 2018

EXAMPLE

The triangle f(n, k) begins:

n\k       0        1        2        3        4       5       6      7     8   9 10

------------------------------------------------------------------------------------

0:        1

1:        1        2

2:        2        6        3

3:        6       22       18        4

4:       24      100      105       40        5

5:      120      548      675      340       75       6

6:      720     3528     4872     2940      875     126       7

7:     5040    26136    39396    27076     9800    1932     196      8

8:    40320   219168   354372   269136   112245   27216    3822    288     9

9:   362880  2053152  3518100  2894720  1346625  379638   66150   6960   405  10

10: 3628800 21257280 38260728 33638000 17084650 5412330 1104411 145200 11880 550 11

... - Wolfdieter Lang, Nov 23 2018

MAPLE

A028421 := proc(n, k) (-1)^(n+k)*(k+1)*Stirling1(n+1, k+1) end: seq(seq(A028421(n, k), k=0..n), n=0..8); # (* Johannes W. Meijer, Oct 07 2009, Revised Sep 09 2012 *)

MATHEMATICA

f[n_, k_] = (k + 1) StirlingS1[n + 1, k + 1] // Abs; Flatten[Table[f[n, k], {n, 0, 9}, {k, 0, n}]][[1 ;; 47]] (* Jean-Fran├žois Alcover, Jun 01 2011, after formula *)

PROG

(Sage) # The function riordan_square is defined in A321620, see comment.

riordan_square(-ln(1 - x), 10, true) # Peter Luschny, Jan 03 2019

CROSSREFS

Row sums give A000254(n+1), n >= 0.

Cf. A132393 (unsigned Stirling1), A061356, A139526, A321620.

From Johannes W. Meijer, Oct 07 2009: (Start)

A000142, A052517, 3*A000399, 5*A000482 are the first four left hand columns; A000027, A002411 are the first two right hand columns.

The asymptotic expansion of E(x,m=2,n) leads to A000254 (n=1), A001705 (n=2), A001711 (n=3), A001716 (n=4), A001721 (n=5), A051524 (n=6), A051545 (n=7), A051560 (n=8), A051562 (n=9), A051564 (n=10), A093905 (triangle) and A165674 (triangle).

Cf. A163931 (E(x,m,n)), A130534 (m=1), A163932 (m=3), A163934 (m=4), A074246 (E(x,m=2,n+1)). (End)

Sequence in context: A227608 A276484 A100641 * A263003 A081745 A240578

Adjacent sequences:  A028418 A028419 A028420 * A028422 A028423 A028424

KEYWORD

tabl,nonn

AUTHOR

Peter Wiggen (wiggen(AT)math.psu.edu)

EXTENSIONS

Edited by Wolfdieter Lang, Nov 23 2018

STATUS

approved

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Last modified March 26 14:18 EDT 2019. Contains 321497 sequences. (Running on oeis4.)