A163936 -id:A163936 - cp's OEIS frontend

A112008 Fourth diagonal of second-order Eulerian triangle A008517. Fourth column (m=3) of triangle A112007.

1, 52, 1452, 32120, 644020, 12440064, 238904904, 4642163952, 92199790224, 1883079661824, 39689578055808, 865023253219584, 19515249341231616, 455924361142656000, 11030149104146035200, 276260563641659673600

Offset: 0

Wolfdieter Lang, Sep 12 2005

			1452= a(2) = 9*52 + 3*328.

Cf. A002539 (3rd diagonal of A008517; third column of A112007).
Contribution from Johannes W. Meijer, Oct 16 2009: (Start)
Equals fourth left hand column of A163936.
(End)

a(n)=A112007(n+3, 3), n>=0.
a(n)= (n+7)*a(n-1) + (n+1)*A002539(n+1), n>=1, a(0)=1.
Contribution from Johannes W. Meijer, Oct 16 2009: (Start)
a(n) = sum((-1)^(n+k)*binomial(2*n+9,k)*stirling1(n-k+8,4-k), k=0..3)
(End)

A163937 Triangle related to the o.g.f.s. of the right-hand columns of A028421 (E(x,m=2,n)).

Original entry on oeis.org

1, 1, 2, 2, 10, 3, 6, 52, 43, 4, 24, 308, 472, 136, 5, 120, 2088, 4980, 2832, 369, 6, 720, 16056, 53988, 49808, 13638, 918, 7, 5040, 138528, 616212, 826160, 381370, 57540, 2167, 8, 40320, 1327392, 7472952, 13570336, 9351260, 2469300, 222908, 4948, 9

Offset: 1

Johannes W. Meijer, Aug 13 2009

The asymptotic expansions of the higher-order exponential integral E(x,m=2,n) lead to triangle A028421, see A163931 for information on E(x,m,n). The o.g.f.s. of the right-hand columns of triangle A028421 have a nice structure: gf(p) = W2(z,p)/(1-z)^(2*p) with p = 1 for the first right-hand column, p = 2 for the second right-hand column, etc. The coefficients of the W2(z,p) polynomials lead to the triangle given above, n >= 1 and 1 <= m <= n. The row sums of this triangle lead to A001147 (minus a(0)), see A163936 for more information.

			The first few W2(z,p) polynomials are
W2(z,p=1) = 1/(1-z)^2;
W2(z,p=2) = (1 +  2*z)/(1-z)^4;
W2(z,p=3) = (2 + 10*z +  3*z^2)/(1-z)^6;
W2(z,p=4) = (6 + 52*z + 43*z^2 + 4*z^3)/(1-z)^8.

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

Row sums equal A001147 (n>=1).
A000142, 2*A001705, are the first two left hand columns.
A000027 is the first right hand column.
Cf. A163931 (E(x,m,n)) and A028421.
Cf. A163936 (E(x,m=1,n)), A163938 (E(x,m=3,n)) and A163939 (E(x,m=4,n)).

with(combinat): a := proc(n, m): add((-1)^(n+k+1)*((m-k)/1!)*binomial(2*n, k)*stirling1(m+n-k-1, m-k), k=0..m-1) end: seq(seq(a(n, m), m=1..n), n=1..9);  # Johannes W. Meijer, revised Nov 27 2012

Table[Sum[(-1)^(n + k + 1)*((m - k)/1!)*Binomial[2*n, k]*StirlingS1[m + n - k - 1, m - k], {k, 0, m - 1}], {n, 1, 10}, {m, 1, n}] // Flatten (* G. C. Greubel, Aug 13 2017 *)

for(n=1,10, for(m=1,n, print1(sum(k=0,m-1, (-1)^(n+k+1)*((m-k)/1!)*binomial(2*n,k) *stirling1(m+n-k-1,m-k)), ", "))) \\ G. C. Greubel, Aug 13 2017

a(n,m) = Sum_{k=0..(m-1)} (-1)^(n+k+1)*((m-k)/1!)*binomial(2*n,k)*Stirling1(m+n-k-1,m-k), 1 <= m <= n.

A112485 Fifth diagonal of second-order Eulerian triangle A008517. Fifth column (m=4) of triangle A112007.

Original entry on oeis.org

1, 114, 5610, 195800, 5765500, 155357384, 4002695088, 101180433024, 2549865473424, 64728375139872, 1666424486271456, 43708768764064128, 1171582385481357696, 32157753536587053312, 905080567903692754176

Offset: 0

Wolfdieter Lang, Sep 12 2005

			5610= a(2) = 11*114 + 3*1452.

Cf. A112008 (fourth diagonal of A008517 and fourth column of A112007).
Contribution from Johannes W. Meijer, Oct 16 2009: (Start)
Equals fifth left hand column of A163936.
(End)

a(n)=A112007(n+4, 4), n>=0.
a(n)= (n+9)*a(n-1) + (n+1)*A112008(n), n>=1, a(0)=1.
Contribution from Johannes W. Meijer, Oct 16 2009: (Start)
a(n) = sum((-1)^(n+k+1)*binomial(2*n+11,k)*stirling1(n-k+10,5-k),k=0..4)
(End)

A288874 Row reversed version of triangle A201637 (second-order Eulerian triangle).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 8, 1, 0, 24, 58, 22, 1, 0, 120, 444, 328, 52, 1, 0, 720, 3708, 4400, 1452, 114, 1, 0, 5040, 33984, 58140, 32120, 5610, 240, 1, 0, 40320, 341136, 785304, 644020, 195800, 19950, 494, 1, 0, 362880, 3733920, 11026296, 12440064, 5765500, 1062500, 67260, 1004, 1, 0, 3628800, 44339040, 162186912, 238904904, 155357384, 44765000, 5326160, 218848, 2026, 1

Offset: 0

Wolfdieter Lang, Jul 20 2017

See A201637, and also A008517 (offset 1 for rows and columns).
The row polynomials of this triangle P(n, x) = Sum_{m=0..n} T(n, m)*x^m appear as numerator polynomials in the o.g.f.s for the diagonal sequences of triangle A132393 (|Stirling1| with offset 0 for rows and columns). See the comment and the P. Bala link there.
For similar triangles see also A112007 and A163936.

			The triangle T(n, m) begins:
n\m 0      1       2        3        4       5       6     7    8  9 ...
0:  1
1:  0      1
2:  0      2       1
3:  0      6       8        1
4:  0     24      58       22        1
5:  0    120     444      328       52       1
6:  0    720    3708     4400     1452     114       1
7:  0   5040   33984    58140    32120    5610     240     1
8:  0  40320  341136   785304   644020  195800   19950   494    1
9:  0 362880 3733920 11026296 12440064 5765500 1062500 67260 1004  1
...

Seiichi Manyama, Rows n = 0..139, flattened
Andrew Elvey Price, Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
Wikipedia, Eulerian numbers of the second kind

Cf. A201637, A008517, A112007, A163936.
Columns m = 0..5: A000007, A000142, A002538, A002539, A112008, A112485.
Diagonals d = 0..3: A000012, A005803, A004301, A006260.
T(2n,n) gives A290306.

T:= (n, k)-> combinat[eulerian2](n, n-k):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Jul 26 2017
# Using the e.g.f:
alias(W = LambertW): len := 10:
egf := (t - 1)*(1/(W(-exp(((t - 1)^2*x - 1)/t)/t) + 1) - 1):
ser := simplify(subs(W(-exp(-1/t)/t) = (-1/t), series(egf, x, len+1))):
seq(seq(n!*coeff(coeff(ser, x, n), t, k), k = 0..n), n = 0..len);  # Peter Luschny, Mar 13 2025

Table[Boole[n == 0] + Sum[(-1)^(n + k) * Binomial[2 n + 1, k] StirlingS1[2 n - m - k, n - m - k], {k, 0, n - m - 1}], {n, 0, 10}, {m, n, 0, -1}] // Flatten (* Michael De Vlieger, Jul 21 2017, after Jean-François Alcover at A201637 *)

T(n, m) = A201637(n, n-m), n >= m >= 0.
Recurrence: T(0, 0) = 1, T(n, -1) = 0, T(n, m) = 0 if n < m, (n-m+1)*T(n-1, m-1) + (n-1+m)*T(n-1, m), n >= 1, m = 0..n; from the A008517 recurrence.
T(0, 0) = 1, T(n, m) = Sum_{p = 0..m-1} (-1)^(n-p)*binomial(2*n+1, p)*A132393(n+m-p, m-p), n >= 1, m = 0..n; from a A008517 program.
T(n, k) = n! * [t^k][x^n] (t - 1)*(1/(W(-exp(((t - 1)^2*x - 1)/t)/t) + 1) - 1) where after expansion W(-exp(-1/t)/t) is substituted by (-1/t). [Inspired by a formula of Shamil Shakirov in A008517.] - Peter Luschny, Mar 13 2025

A290306 Number of permutations of the multiset {1,1,2,2,...,2n,2n} having exactly n ascents and no number smaller than k between the two occurrences of any number k.

Original entry on oeis.org

1, 2, 58, 4400, 644020, 155357384, 56041398784, 28299910066112, 19076135772884080, 16558710676700081120, 17997592513561138205728, 23948993629880321407298816, 38303802347672648465676584704, 72510806370598644118983905976320, 160368191672482402606757066578885120

Offset: 0

Alois P. Heinz, Jul 26 2017

nonn

			a(1) = 2: 1122, 1221.
a(2) = 58: 11224433, 11244332, 11332244, 11332442, 11334422, 11344322, ..., 44112233, 44112332, 44122133, 44122331, 44123321, 44133122.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 270.

Alois P. Heinz, Table of n, a(n) for n = 0..206
Wikipedia, Eulerian numbers of the second kind

Cf. A008517, A201637, A112007, A163936, A288874.

a:= n-> combinat[eulerian2](2*n, n):
seq(a(n), n=0..20);
# second Maple program:
b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
     `if`(n=0, 1, (2*n-k-1)*b(n-1, k-1)+(k+1)*b(n-1, k)))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..20);

b[n_, k_]:=b[n, k]=If[k<0 || k>n, 0, If[n==0, 1, (2*n - k  - 1)*b[n - 1, k - 1] + (k + 1)*b[n - 1, k]]]; Table[b[2n, n], {n, 0, 20}] (* Indranil Ghosh, Jul 27 2017, after second Maple program *)
Flatten[{1, Table[Sum[(-1)^(n-k) * Binomial[4*n + 1, n - k] * StirlingS1[2*n + k, k], {k, 1, n}], {n, 1, 15}]}] (* Vaclav Kotesovec, Aug 11 2018 *)

a(n) = A201637(2n,n) = A288874(2n,n) = <<2n,n>>, with <<.,.>> = second order Eulerian numbers or Eulerian numbers of the second kind.

a(n) ~ c * d^n * n^(2*n - 1/2), where d = 1.6899458441572699524424834032837129180107588318196320162637478870996171397... and c = 3.5414537300298411499842602111667139605122817390785452902057395704515855797... - Vaclav Kotesovec, Aug 11 2018

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A112008 Fourth diagonal of second-order Eulerian triangle A008517. Fourth column (m=3) of triangle A112007.

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A163937 Triangle related to the o.g.f.s. of the right-hand columns of A028421 (E(x,m=2,n)).

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A112485 Fifth diagonal of second-order Eulerian triangle A008517. Fifth column (m=4) of triangle A112007.

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A288874 Row reversed version of triangle A201637 (second-order Eulerian triangle).

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A290306 Number of permutations of the multiset {1,1,2,2,...,2n,2n} having exactly n ascents and no number smaller than k between the two occurrences of any number k.

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