A355609 -id:A355609

Search: a355609 -id:a355609

Displaying 1-4 of 4 results found.

page 1

A355607

Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 + x)^(x^k).

+10
6

1, 1, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, -3, 0, 1, 0, 0, 6, 20, 0, 1, 0, 0, 0, -12, -90, 0, 1, 0, 0, 0, 24, 40, 594, 0, 1, 0, 0, 0, 0, -60, 180, -4200, 0, 1, 0, 0, 0, 0, 120, 240, -1512, 34544, 0, 1, 0, 0, 0, 0, 0, -360, -1260, 11760, -316008, 0, 1, 0, 0, 0, 0, 0, 720, 1680, 28224, -38880, 3207240, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,9

LINKS

Seiichi Manyama, Antidiagonals n = 0..139, flattened

FORMULA

T(0,k) = 1 and T(n,k) = -(n-1)! * Sum_{j=k+1..n} (-1)^(j-k) * j/(j-k) * T(n-j,k)/(n-j)! for n > 0.

T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling1(n-k*j,j)/(n-k*j)!.

EXAMPLE

Square array begins:

1, 1, 1, 1, 1, 1, 1, ...

1, 0, 0, 0, 0, 0, 0, ...

0, 2, 0, 0, 0, 0, 0, ...

0, -3, 6, 0, 0, 0, 0, ...

0, 20, -12, 24, 0, 0, 0, ...

0, -90, 40, -60, 120, 0, 0, ...

0, 594, 180, 240, -360, 720, 0, ...

PROG

(PARI) T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 1)/(n-k*j)!);

CROSSREFS

Columns k=1..4 give A007113, A007121, (-1)^n * A353229(n), A354625.

Cf. A292892, A355609, A355619.

KEYWORD

sign,tabl,look

AUTHOR

Seiichi Manyama, Jul 09 2022

STATUS

approved

A355610

Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - x)^(-x^k/k!).

+10
6

1, 1, 1, 1, 0, 2, 1, 0, 2, 6, 1, 0, 0, 3, 24, 1, 0, 0, 3, 20, 120, 1, 0, 0, 0, 6, 90, 720, 1, 0, 0, 0, 4, 20, 594, 5040, 1, 0, 0, 0, 0, 10, 180, 4200, 40320, 1, 0, 0, 0, 0, 5, 40, 1134, 34544, 362880, 1, 0, 0, 0, 0, 0, 15, 210, 7980, 316008, 3628800, 1, 0, 0, 0, 0, 0, 6, 70, 1904, 71280, 3207240, 39916800

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

Seiichi Manyama, Antidiagonals n = 0..139, flattened

FORMULA

T(0,k) = 1 and T(n,k) = (n-1)!/k! * Sum_{j=k+1..n} j/(j-k) * T(n-j,k)/(n-j)! for n > 0.

T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} |Stirling1(n-k*j,j)|/(k!^j * (n-k*j)!).

EXAMPLE

Square array begins:

1, 1, 1, 1, 1, 1, 1, ...

1, 0, 0, 0, 0, 0, 0, ...

2, 2, 0, 0, 0, 0, 0, ...

6, 3, 3, 0, 0, 0, 0, ...

24, 20, 6, 4, 0, 0, 0, ...

120, 90, 20, 10, 5, 0, 0, ...

720, 594, 180, 40, 15, 6, 0, ...

PROG

(PARI) T(n, k) = n!*sum(j=0, n\(k+1), abs(stirling(n-k*j, j, 1))/(k!^j*(n-k*j)!));

CROSSREFS

Columns k=0..4 give A000142, A066166, A351492, A351493, A355507.

Cf. A355609, A355619.

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Jul 09 2022

STATUS

approved

A354624

Expansion of e.g.f. (1 - x)^(-x^4).

+10
2

1, 0, 0, 0, 0, 120, 360, 1680, 10080, 72576, 2419200, 25660800, 279417600, 3286483200, 41894012160, 794511244800, 13755488947200, 238514695372800, 4269265386946560, 79696849513881600, 1658065431859200000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

Table of n, a(n) for n=0..20.

FORMULA

a(0) = 1; a(n) = (n-1)! * Sum_{k=5..n} k/(k-4) * a(n-k)/(n-k)!.

a(n) = n! * Sum_{k=0..floor(n/5)} |Stirling1(n-4*k,k)|/(n-4*k)!.

a(n) ~ n! * (1 - 4/n - 16*log(n)/n^2). - Vaclav Kotesovec, Jul 21 2022

PROG

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-x^4)))

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x^4*log(1-x))))

(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=5, i, j/(j-4)*v[i-j+1]/(i-j)!)); v;

(PARI) a(n) = n!*sum(k=0, n\5, abs(stirling(n-4*k, k, 1))/(n-4*k)!);

CROSSREFS

Column k=4 of A355609.

Cf. A353228, A353229.

Cf. A354625.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Jul 09 2022

STATUS

approved

A355665

Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + x^k * log(1 - x)).

+10
1

1, 1, 1, 1, 0, 3, 1, 0, 2, 14, 1, 0, 0, 3, 88, 1, 0, 0, 6, 32, 694, 1, 0, 0, 0, 12, 150, 6578, 1, 0, 0, 0, 24, 40, 1524, 72792, 1, 0, 0, 0, 0, 60, 900, 12600, 920904, 1, 0, 0, 0, 0, 120, 240, 6048, 147328, 13109088, 1, 0, 0, 0, 0, 0, 360, 1260, 43680, 1705536, 207360912

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

Table of n, a(n) for n=0..65.

FORMULA

T(0,k) = 1 and T(n,k) = n! * Sum_{j=k+1..n} 1/(j-k) * T(n-j,k)/(n-j)! for n > 0.

T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} j! * |Stirling1(n-k*j,j)|/(n-k*j)!.

EXAMPLE

Square array begins:

1, 1, 1, 1, 1, 1, 1, ...

1, 0, 0, 0, 0, 0, 0, ...

3, 2, 0, 0, 0, 0, 0, ...

14, 3, 6, 0, 0, 0, 0, ...

88, 32, 12, 24, 0, 0, 0, ...

694, 150, 40, 60, 120, 0, 0, ...

6578, 1524, 900, 240, 360, 720, 0, ...

PROG

(PARI) T(n, k) = n!*sum(j=0, n\(k+1), j!*abs(stirling(n-k*j, j, 1))/(n-k*j)!);

CROSSREFS

Columns k=0..3 give A007840, A052830, A351503, A351504.

Cf. A355609, A355652.

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Jul 13 2022

STATUS

approved

page 1

Search completed in 0.005 seconds