A008517 -id:A008517 - cp's OEIS frontend

A112500 Triangle of column sequences with a certain o.g.f. pattern.

1, 1, 1, 1, 4, 1, 1, 11, 10, 1, 1, 26, 60, 20, 1, 1, 57, 282, 225, 35, 1, 1, 120, 1149, 1882, 665, 56, 1, 1, 247, 4272, 13070, 9107, 1666, 84, 1, 1, 502, 14932, 79872, 100751, 35028, 3696, 120, 1, 1, 1013, 49996, 444902, 957197, 584325, 113428, 7470, 165, 1, 1

Offset: 0

Wolfdieter Lang, Oct 14 2005

The column o.g.f.s of this triangle appear as factors in the column o.g.f.s of triangle A008517 (second-order Eulerian numbers).

			Rows: [1]; [1,1]; [1,4,1]; [1,11,10,1]; [1,26,60,20,1]; [1,57,282,225,35,1]; ...
a(4,3)= 60 = 6 + 12 + 9 + 12 + 9 + 12 from the binomial(4,2)=6 terms of the sum corresponding to (n_1,n_2,n_3) = (2,0,0), (0,2,0), (0,0,2), (1,1,0), (1,0,1) and (0,1,1).

W. Lang, First ten rows.

Cf. A112501 (row sums).

G.f. column k: G(k, x):= x^(k-1)/product((1-j*x)^(k-j+1), j=1..k), k>=1.
The column sequences start with A000012 (powers of 1), A000295 (Eulerian numbers), A112502-A112504.
a(n+k-1, k)=sum of product(binomial(n_j + k - 1, k - 1)*j^(n_j), j=1..k) with sum(n_j, j=1..k)=n, n_j >=0. There are binomial(n+k-1, k-1) terms of this sum and 1<=k<=n+1. a(n, k)=0 if n+1

A112494 Sixth diagonal of the Stirling2 triangle A048993 and sixth column of triangle A008278.

Original entry on oeis.org

1, 63, 966, 7770, 42525, 179487, 627396, 1899612, 5135130, 12662650, 28936908, 62022324, 125854638, 243577530, 452329200, 809944464, 1404142047, 2364885369, 3880739170, 6220194750, 9759104355, 15015551265, 22693687380, 33738295500, 49402080000, 71327958156

Offset: 6

Wolfdieter Lang, Oct 14 2005

T. D. Noe, Table of n, a(n) for n = 6..1000
Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq. 13 (2010), 10.4.4, page 5.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 29.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A008517, A112493.

Cf. A001298 (fifth diagonal, resp. column).

Table[StirlingS2[n, n-5], {n, 6, 100}] (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)

for(n=6,50, print1(stirling(n,n-5,2), ", ")) \\ G. C. Greubel, Oct 22 2017

Vec(x^6*(1 + 52*x + 328*x^2 + 444*x^3 + 120*x^4) / (1 - x)^11 + O(x^40)) \\ Colin Barker, Nov 04 2017

[stirling_number2(n,n-5) for n in range(6, 30)] # Zerinvary Lajos, May 16 2009

a(n) = Stirling2(n, n-5) with Stirling2(n, m)=A048993(n, m). a(n) = A008278(n+5, 6).
a(n) = sum(A008517(5, m+1)*binomial(n+5-m, 2*5), m=0..4) from the o.g.f. See p. 257 eq. (6.43) of the R. L. Graham et al. book quoted in A008517.
O.g.f.: x*sum(A008517(5, m+1)*x^m, m=0..4)/(1-x)^11 with the fifth row [1, 52, 328, 444, 120] of the second-order Eulerian triangle A008517.
E.g.f. with offset n=-4: exp(x)*sum(A112493(5, m)*(x^(m+5))/(m+5)!, m=0..5) with the k=5 row [1, 57, 546, 1750, 2205, 945] of triangle A112493.
a(n) = sum(A112493(5, m)*binomial(n+4, 5+m), m=0..5) from the e.g.f. (coefficients from A112493(5, m) are [1, 57, 546, 1750, 2205, 945]).
With an offset of 1 the o.g.f. is D^5(x/(1-x)), where D is the operator x/(1-x)*d/dx. - Peter Bala, Jul 02 2012
G.f.: x^6*(1 + 52*x + 328*x^2 + 444*x^3 + 120*x^4) / (1 - x)^11. - Colin Barker, Nov 04 2017

A157011 Triangle T(n,k) read by rows: T(n,k) = (k-1)T(n-1,k) + (n-k+2)T(n-1, k-1), with T(n,1)=1, for 1 <= k <= n, n >= 1.

Original entry on oeis.org

1, 1, 2, 1, 5, 4, 1, 9, 23, 8, 1, 14, 82, 93, 16, 1, 20, 234, 607, 343, 32, 1, 27, 588, 2991, 3800, 1189, 64, 1, 35, 1365, 12501, 30155, 21145, 3951, 128, 1, 44, 3010, 47058, 195626, 256500, 108286, 12749, 256, 1, 54, 6416, 165254, 1111910, 2456256, 1932216, 522387, 40295, 512

Offset: 1

Roger L. Bagula, Feb 21 2009

Row sums are apparently in A002627.

The Mathematica code gives ten sequences of which the first few are in the OEIS (see Crossrefs section). - G. C. Greubel, Feb 22 2019

			The triangle starts in row n=1 as:
  1;
  1,  2;
  1,  5,    4;
  1,  9,   23,      8;
  1, 14,   82,     93,      16;
  1, 20,  234,    607,     343,      32;
  1, 27,  588,   2991,    3800,    1189,      64;
  1, 35, 1365,  12501,   30155,   21145,    3951,    128;
  1, 44, 3010,  47058,  195626,  256500,  108286,  12749,   256;
  1, 54, 6416, 165254, 1111910, 2456256, 1932216, 522387, 40295, 512;

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A000096 (column k=1), A002627, A008517.

Cf. This sequence (m=0), A008292 (m=1), A157012 (m=2), A157013 (m=3).

A157011 := proc(n,k) if k <0 or k >= n then 0; elif k  =0 then 1; else k*procname(n-1,k)+(n-k+1)*procname(n-1,k-1) ; end if; end proc: # R. J. Mathar, Jun 18 2011

e[n_, 0, m_]:= 1;
e[n_, k_, m_]:= 0 /; k >= n;
e[n_, k_, m_]:= (k+m)*e[n-1, k, m] + (n-k+1-m)*e[n-1, k-1, m];
Table[Flatten[Table[Table[e[n, k, m], {k,0,n-1}], {n,1,10}]], {m,0,10}]
T[n_, 1]:= 1; T[n_, n_]:= 2^(n-1); T[n_, k_]:= T[n, k] = (k-1)*T[n-1, k] + (n-k+2)*T[n-1, k-1]; Table[T[n, k], {n, 1, 10}, {k, 1, n}]//Flatten (* G. C. Greubel, Feb 22 2019 *)

{T(n, k) = if(k==1, 1, if(k==n, 2^(n-1), (k-1)*T(n-1, k) + (n-k+2)* T(n-1, k-1)))};
for(n=1, 10, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Feb 22 2019

def T(n, k):
    if (k==1):
        return 1
    elif (k==n):
        return 2^(n-1)
    else: return (k-1)*T(n-1, k) + (n-k+2)* T(n-1, k-1)
[[T(n, k) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Feb 22 2019

A112002 Seventh diagonal of triangle A008275 (Stirling1) and seventh column of |A008276|.

Original entry on oeis.org

720, 13068, 118124, 723680, 3416930, 13339535, 44990231, 135036473, 368411615, 928095740, 2185031420, 4853222764, 10246937272, 20692933630, 40171771630, 75289668850, 136717357942, 241276443496, 414908513800, 696829576300

Offset: 1

Wolfdieter Lang, Sep 12 2005

T. D. Noe, Table of n, a(n) for n = 1..1000

Sixth diagonal A053567; A130534.

[StirlingFirst(n+6, n): n in [1..20]]; // Vincenzo Librandi, Aug 09 2015

A112002 := proc(n) combinat[stirling1](n+6,n) ; end proc: # R. J. Mathar, Jun 08 2011

Table[StirlingS1[n+6, n], {n, 1, 20}] (* Jean-François Alcover, Mar 05 2014 *)

[stirling_number1(n,n-6) for n in range(7, 27)] # Zerinvary Lajos, May 16 2009

a(n)= Stirling1(n+6, n), n>=1, with Stirling1(n, k)= A008275(n, k).
E.g.f. with offset 6: exp(x)*sum(A112486(6, m)*(x^(6+m))/(6+m)!, m=0..6).
a(n)= (f(n+5, 6)/12!)*sum(A112486(6, m)*f(12, 6-m)*f(n-1, m), m=0..min(6, n-1)), with the falling factorials f(n, k):=n*(n-1)*...*(n-(k-1)). From the e.g.f.
a(n)=(binomial(n+6, 7)/r(8, 5))*sum(A112007(5, m)*r(n+7, 5-m)*f(n-1, m), m=0..5), with rising factorials r(n, k):=n*(n+1)*...*(n+(k-1)) and falling factorials f(n, m). From the g.f.
G.f.: x*(720+3708*x+4400*x^2+1452*x^3+114*x^4+x^5)/(1-x)^13. See row k=5 of triangles A112007 or A008517 for the coefficients.
Explicit formula: a(n) = n(n + 1)(n + 2)(n + 3)(n + 4)(n + 5)(n + 6)(63n^5 + 1575n^4 + 15435n^3 + 73801n^2 + 171150n + 152696)/2903040. - Vaclav Kotesovec, Jan 30 2010

A144969 Stirling numbers of second kind S(n,n-6).

Original entry on oeis.org

0, 1, 127, 3025, 34105, 246730, 1323652, 5715424, 20912320, 67128490, 193754990, 512060978, 1256328866, 2892439160, 6302524580, 13087462580, 26046574004, 49916988803, 92484925445, 166218969675, 290622864675, 495564056130, 825906183960, 1347860993700

Offset: 6

Vladimir Joseph Stephan Orlovsky, Sep 27 2008

T. D. Noe, Table of n, a(n) for n = 6..1000
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 29.
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A008517, A112493.

Table[StirlingS2[n,n-6], {n,6,30}] (* Harvey P. Dale, Sep 21 2011 *)

concat(0, Vec(x^7*(720*x^5 +3708*x^4 +4400*x^3 +1452*x^2 +114*x +1 )/(1-x)^13 + O(x^100))) \\ Colin Barker, Oct 28 2014

for(n=6,50, print1(stirling(n,n-6,2), ", ")) \\ G. C. Greubel, Oct 23 2017

[stirling_number2(n,n-6) for n in range(6, 28)] # Zerinvary Lajos, May 16 2009

With an offset of 1 the o.g.f. is D^6(x/(1-x)), where D is the operator x/(1-x)*d/dx. See A008517. For the e.g.f. see A112493. - Peter Bala, Jul 02 2012

G.f.: x^7*(720*x^5 +3708*x^4 +4400*x^3 +1452*x^2 +114*x +1)/(1-x-)^13. - Colin Barker, Oct 28 2014

A157012 Riordan's general Eulerian recursion: T(n,k) = (k+2)T(n-1, k) + (n-k) T(n-1, k-1), with T(n,0) = 1, T(n,n) = 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 18, 14, 1, 0, 1, 58, 110, 33, 1, 0, 1, 179, 672, 495, 72, 1, 0, 1, 543, 3583, 5163, 1917, 151, 1, 0, 1, 1636, 17590, 43730, 32154, 6808, 310, 1, 0, 1, 4916, 81812, 324190, 411574, 176272, 22904, 629, 1, 0

Offset: 0

Roger L. Bagula, Feb 21 2009

Row sums are:
{1, 1, 2, 7, 34, 203, 1420, 11359, 102230, 1022299,...}.
This recursion set doesn't seem to produce the Eulerian 2nd A008517.
The Mathematica code gives ten sequences of which the first few are in the OEIS (see Crossrefs section). - G. C. Greubel, Feb 22 2019

			Triangle begins with:
1.
1,    0.
1,    1,     0.
1,    5,     1,      0.
1,   18,    14,      1,      0.
1,   58,   110,     33,      1,      0.
1,  179,   672,    495,     72,      1,     0.
1,  543,  3583,   5163,   1917,    151,     1,   0.
1, 1636, 17590,  43730,  32154,   6808,   310,   1,   0.
1, 4916, 81812, 324190, 411574, 176272, 22904, 629,   1,   0.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 214-215

G. C. Greubel, Rows n=0..100 of triangle, flattened

Cf. A008517.

Cf. A157011 (m=0), A008292 (m=1), This Sequence (m=2), A157013 (m=3).

e[n_, 0, m_]:= 1;
e[n_, k_, m_]:= 0 /; k >= n;
e[n_, k_, m_]:= (k+m)*e[n-1, k, m] +(n-k+1-m)*e[n-1, k-1, m];
Table[Flatten[Table[Table[e[n, k, m], {k,0,n-1}], {n,1,10}]], {m,0,10}]
T[n_, 0]:= 1; T[n_, n_]:= 0; T[n_, k_]:= T[n, k] = (k+2)*T[n-1, k] +(n-k) *T[n-1, k-1]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 22 2019 *)

{T(n, k) = if(k==0, 1, if(k==n, 0, (k+2)*T(n-1, k) + (n-k)* T(n-1, k-1)))};
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Feb 22 2019

def T(n, k):
    if (k==0): return 1
    elif (k==n): return 0
    else: return (k+2)*T(n-1, k) + (n-k)* T(n-1, k-1)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 22 2019

e(n,k,m) = (k+m)*e(n-1, k, m) + (n-k+1-m)*e(n-1, k-1, m) for m=2.

T(n,k) = (k+2)*T(n-1, k) + (n-k)*T(n-1, k-1), with T(n,0) = 1, T(n,n) = 0. - G. C. Greubel, Feb 22 2019

A157013 Riordan's general Eulerian recursion: T(n, k) = (k+2)T(n-1, k) + (n-k-1) T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-1)^(n-1).

Original entry on oeis.org

1, 1, -1, 1, -4, 1, 1, -15, 5, -1, 1, -58, 10, -6, 1, 1, -229, -66, -26, 7, -1, 1, -912, -1017, -288, 23, -8, 1, 1, -3643, -8733, -4779, -415, -41, 9, -1, 1, -14566, -61880, -63606, -17242, -1158, 40, -10, 1, 1, -58257, -396796, -691036, -375118, -60990, -1956, -60, 11, -1

Offset: 1

Roger L. Bagula, Feb 21 2009

Row sums are {1, 0, -2, -10, -52, -314, -2200, -17602, -158420, -1584202, ...}.
This recursion set doesn't seem to produce the Eulerian 2nd A008517.
The Mathematica code gives ten sequences of which the first few are in the OEIS (see Crossrefs section). - G. C. Greubel, Feb 22 2019

			Triangle begins with:
1.
1,     -1.
1,     -4,       1.
1,    -15,       5,      -1.
1,    -58,      10,      -6,       1.
1,   -229,     -66,     -26,       7,     -1.
1,   -912,   -1017,    -288,      23,     -8,     1.
1,  -3643,   -8733,   -4779,    -415,    -41,     9,   -1.
1, -14566,  -61880,  -63606,  -17242,  -1158,    40,  -10,   1.
1, -58257, -396796, -691036, -375118, -60990, -1956,  -60,  11,  -1.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 214-215

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A008517.

Cf. A157011 (m=0), A008292 (m=1), A157012 (m=2), this sequence (m=3).

e[n_, 0, m_]:= 1;
e[n_, k_, m_]:= 0 /; k >= n;
e[n_, k_, m_]:= (k+m)*e[n-1, k, m] + (n-k+1-m)*e[n-1, k-1, m];
Table[Flatten[Table[Table[e[n, k, m], {k,0,n-1}], {n,1,10}]], {m,0,10}]
T[n_,1]:=1; T[n_,n_]:=(-1)^(n-1); T[n_,k_]:= T[n,k] = (k+2)*T[n-1,k] + (n-k-1)*T[n-1,k-1]; Table[T[n,k], {n,1,10}, {k,1,n}]//Flatten (* G. C. Greubel, Feb 22 2019 *)

{T(n, k) = if(k==1, 1, if(k==n, (-1)^(n-1), (k+2)*T(n-1, k) + (n-k-1)* T(n-1, k-1)))};
for(n=1, 10, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Feb 22 2019

def T(n, k):
    if (k==1): return 1
    elif (k==n): return (-1)^(n-1)
    else: return (k+2)*T(n-1, k) + (n-k-1)* T(n-1, k-1)
[[T(n, k) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Feb 22 2019

e(n,k,m)= (k+m)*e(n-1, k, m) + (n-k+1-m)*e(n-1, k-1, m) with m=3.

T(n, k) = (k+2)*T(n-1, k) + (n-k-1)*T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-1)^(n-1). - G. C. Greubel, Feb 22 2019

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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cp's OEIS Frontend

A006260 Second-order Eulerian numbers <>.

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A112500 Triangle of column sequences with a certain o.g.f. pattern.

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A112494 Sixth diagonal of the Stirling2 triangle A048993 and sixth column of triangle A008278.

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A157011 Triangle T(n,k) read by rows: T(n,k) = (k-1)*T(n-1,k) + (n-k+2)*T(n-1, k-1), with T(n,1)=1, for 1 <= k <= n, n >= 1.

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A112002 Seventh diagonal of triangle A008275 (Stirling1) and seventh column of |A008276|.

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A144969 Stirling numbers of second kind S(n,n-6).

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A157012 Riordan's general Eulerian recursion: T(n,k) = (k+2)*T(n-1, k) + (n-k) * T(n-1, k-1), with T(n,0) = 1, T(n,n) = 0.

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A157011 Triangle T(n,k) read by rows: T(n,k) = (k-1)T(n-1,k) + (n-k+2)T(n-1, k-1), with T(n,1)=1, for 1 <= k <= n, n >= 1.

A157012 Riordan's general Eulerian recursion: T(n,k) = (k+2)T(n-1, k) + (n-k) T(n-1, k-1), with T(n,0) = 1, T(n,n) = 0.

A157013 Riordan's general Eulerian recursion: T(n, k) = (k+2)T(n-1, k) + (n-k-1) T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-1)^(n-1).