A177042 -id:A177042 - cp's OEIS frontend

A008518 Triangle of Eulerian numbers with rows multiplied by 1 + x.

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 12, 22, 12, 1, 1, 27, 92, 92, 27, 1, 1, 58, 359, 604, 359, 58, 1, 1, 121, 1311, 3607, 3607, 1311, 121, 1, 1, 248, 4540, 19912, 31238, 19912, 4540, 248, 1, 1, 503, 15110, 102842, 244424, 244424, 102842, 15110, 503, 1

Offset: 0

N. J. A. Sloane

			Triangle begins:
   1;
   1,   1;
   1,   2,    1;
   1,   5,    5,    1;
   1,  12,   22,   12,    1;
   1,  27,   92,   92,   27,    1;
   1,  58,  359,  604,  359,   58,   1;
   1, 121, 1311, 3607, 3607, 1311, 121, 1;
   ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.

Alois P. Heinz, Rows n = 0..140, flattened
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 22.

Cf. A000007, A008292, A098558 (row sums), A177042 (T(2*n, n)).

Columns include A000325 (k=1).

Eulerian:= func< n, k | (&+[(-1)^j*Binomial(n+1, j)*(k-j+1)^n: j in [0..k+1]]) >;
[Eulerian(n, k-1) + Eulerian(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jun 18 2024

t[n_ /; n >= 0, 0] = 1; t[n_, k_] /; k<0 || k>n = 0; t[n_, k_] := t[n, k] = (n-k) t[n-1, k-1] + (k+1) t[n-1, k];
A[n_, k_] /; k == n+1 = 0; A[n_, k_] := t[n, n-k];
T[n_, k_] := A[n, k] + A[n, k+1];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 26 2019, after Franck Maminirina Ramaharo *)

def Eulerian(n,k): return sum((-1)^j*binomial(n+1, j)*(k-j+1)^n for j in range(k+2))
flatten([[Eulerian(n,k-1) + Eulerian(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 18 2024

E.g.f.: (exp(x) - y*exp(y*x))/(exp(y*x) - y*exp(x)). - Vladeta Jovovic, Apr 06 2001
T(n,k) = A123125(n,k) + A123125(n,k+1), with A123125(n,n+1) = 0. - Franck Maminirina Ramaharo, Oct 21 2018
From G. C. Greubel, Jun 18 2024: (Start)
T(n, n-k) = T(n, k).
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n). (End)

More terms from Vladeta Jovovic, Apr 06 2001

A303285 Number of permutations p of [2n] such that the sequence of ascents and descents of p0 forms a Dyck path.

Original entry on oeis.org

1, 1, 8, 172, 7296, 518324, 55717312, 8460090160, 1726791794432, 456440969661508, 151770739970889792, 62022635037246022000, 30564038464166725328768, 17876875858414492985045712, 12245573879235563308351042496, 9711714975145772145881269175104

Offset: 0

Alois P. Heinz, Apr 20 2018

nonn

Here p is a permutation of 1,2,3,...,2n, and p0 refers to the string p followed by 0.
Also the number of permutations p of [2n] such that the sequence of ascents and descents of 0p forms a Dyck path. a(2) = 8: 1432, 2143, 2431, 3142, 3241, 3421, 4132, 4231.
Also the number of permutations p of [2n] that are of odd order and whose M statistic (as defined in the Spiro paper) is equal to n-1. - Sam Spiro, Nov 01 2018

			a(0) = 1: the empty permutation.
a(1) = 1: 12.
a(2) = 8: 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412.

Alois P. Heinz, Table of n, a(n) for n = 0..200
S. Spiro, Ballot Permutations, Odd Order Permutations, and a New Permutation Statistic, arXiv:1810.00993 [math.CO], 2018.
Wikipedia, Counting lattice paths

Bisection (even part) of A303284.
Bisection (even part) of A303287.
Column k=0 of A316728.
Cf. A177042, A180056, A316727, A321280.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t>0,   add(b(u-j, o+j-1, t-1), j=1..u), 0)+
      `if`(o+u>t, add(b(u+j-1, o-j, t+1), j=1..o), 0))
    end:
a:= n-> b(0, 2*n, 0):
seq(a(n), n=0..20);

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t > 0, Sum[b[u - j, o + j - 1, t - 1], {j, 1, u}], 0] + If[o + u > t, Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}], 0]];
a[n_] := b[0, 2n, 0];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 29 2018, from Maple *)

\\ here b(n) is A177042
b(n)={if(n==0, 1, 2*sum(k=0, n, (-1)^k*binomial(2*n+1,k)*(n-k+1)^(2*n)));}
a(n)={if(n==0, 1, sum(k=1, n, binomial(2*n, 2*k-1)*b(k-1)*b(n-k))/2);} \\ Andrew Howroyd, Nov 01 2018

a(n) ~ c * 2^(2*n) * n^(2*n - 1) / exp(2*n), where c = 8.838022110416151362523442920999767406145711133564692... - Vaclav Kotesovec, May 22 2018
a(n) = (1/2)*Sum_{k odd} binomial(2*n,k)*A177042((k-1)/2)*A177042((2n-k-1)/2) for n>0. - Sam Spiro, Nov 01 2018
a(n) = A321280(2n,n-1) for n >= 1. - Alois P. Heinz, Nov 02 2018

A303287 Number of permutations p of [n] such that the sequence of ascents and descents of p or of p0 (if n is even) forms a Dyck path.

Original entry on oeis.org

1, 1, 1, 2, 8, 22, 172, 604, 7296, 31238, 518324, 2620708, 55717312, 325024572, 8460090160, 55942352184, 1726791794432, 12765597850950, 456440969661508, 3730771315561300, 151770739970889792, 1359124435588313876, 62022635037246022000, 603916464771468176392

Offset: 0

Alois P. Heinz, Apr 20 2018

nonn

			a(0) = 1: the empty permutation.
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 2: 132, 231.
a(4) = 8: 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412.
a(5) = 22: 12543, 13254, 13542, 14253, 14352, 14532, 15243, 15342, 23154, 23541, 24153, 24351, 24531, 25143, 25341, 34152, 34251, 34521, 35142, 35241, 45132, 45231.

Alois P. Heinz, Table of n, a(n) for n = 0..400
Wikipedia, Counting lattice paths

Bisections give: A303285 (even part), A177042 (odd part).

Cf. A180056, A304001, A303284, A304777, A304778.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t>0,   add(b(u-j, o+j-1, t-1), j=1..u), 0)+
      `if`(o+u>t, add(b(u+j-1, o-j, t+1), j=1..o), 0))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t > 0, Sum[b[u - j, o + j - 1, t - 1], {j, 1, u}], 0] + If[o + u > t, Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}], 0]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 25 2018, translated from Maple *)

a(2n) = A303284(2n).

A316728 Number T(n,k) of permutations of {0,1,...,2n} with first element k whose sequence of ascents and descents forms a Dyck path; triangle T(n,k), n>=0, 0<=k<=2n, read by rows.

Original entry on oeis.org

1, 1, 1, 0, 8, 7, 5, 2, 0, 172, 150, 121, 87, 52, 22, 0, 7296, 6440, 5464, 4411, 3337, 2306, 1380, 604, 0, 518324, 463578, 405024, 344260, 283073, 223333, 166856, 115250, 69772, 31238, 0, 55717312, 50416894, 44928220, 39348036, 33777456, 28318137, 23068057, 18117190, 13543456, 9409366, 5759740, 2620708, 0

Offset: 0

Alois P. Heinz, Jul 11 2018

			T(2,0) = 8: 01432, 02143, 02431, 03142, 03241, 03421, 04132, 04231.
T(2,1) = 7: 12043, 12430, 13042, 13240, 13420, 14032, 14230.
T(2,2) = 5: 23041, 23140, 23410, 24031, 24130.
T(2,3) = 2: 34021, 34120.
T(2,4) = 0.
Triangle T(n,k) begins:
       1;
       1,      1,      0;
       8,      7,      5,      2,      0;
     172,    150,    121,     87,     52,     22,      0;
    7296,   6440,   5464,   4411,   3337,   2306,   1380,    604,     0;
  518324, 463578, 405024, 344260, 283073, 223333, 166856, 115250, 69772, 31238, 0;

Alois P. Heinz, Rows n = 0..100, flattened
Wikipedia, Counting lattice paths

Column k=0 gives A303285.
Row sums and T(n+1,2n+1) give A177042.
T(n,n) gives A316727.
T(n+1,n) gives A316730.
T(n,2n) gives A000007.
Cf. A079484.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t>0,   add(b(u-j, o+j-1, t-1), j=1..u), 0)+
      `if`(o+u>t, add(b(u+j-1, o-j, t+1), j=1..o), 0))
    end:
T:= (n, k)-> b(k, 2*n-k, 0):
seq(seq(T(n, k), k=0..2*n), n=0..8);

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
     If[t > 0,     Sum[b[u - j, o + j - 1, t - 1], {j, 1, u}], 0] +
     If[o + u > t, Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}], 0]];
T[n_, k_] := b[k, 2n - k, 0];
Table[Table[T[n, k], {k, 0, 2n}], {n, 0, 8}] // Flatten (* Jean-François Alcover, Mar 27 2021, after Alois P. Heinz *)

Sum_{k=0..2n} T(n,k) = T(n+1,2n+1) = A177042(n).

Sum_{k=0..2n} (k+1) * T(n,k) = A079484(n).

A172010 a(n) = 2A142458(2n, n)/(n+1).

Original entry on oeis.org

1, 26, 2741, 683870, 315704418, 234725594388, 257237392999893, 390832857108454838, 787178784737043042806, 2031210797603911366282796, 6536955866068372922068141666, 25676217636579568989377656129516, 120915166829869713032692550819662756, 672580820552232143302651758669053327784

Offset: 1

Roger L. Bagula, Nov 19 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 1..190

Cf. A142458, A177042, A177043.

T[n_, k_, m_]:= T[n,k,m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1,k-1,m] + (m*k-m+ 1)*T[n-1,k,m]];
a[n_]:= 2*T[2*n,n,3]/(n+1);
Table[a[n], {n,30}] (* modified by G. C. Greubel, Mar 14 2022 *)

@CachedFunction
def T(n,k,m): # A142458
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
[2*T(2*n,n,3)/(n+1) for n in (1..30)] # G. C. Greubel, Mar 14 2022

a(n) = 2*A142458(2*n, n)/(n+1).

Name corrected and more terms added by G. C. Greubel, Mar 14 2022

A321280 Number T(n,k) of permutations p of [n] with exactly k descents such that the up-down signature of p has nonnegative partial sums; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/2)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 8, 1, 22, 22, 1, 52, 172, 1, 114, 856, 604, 1, 240, 3488, 7296, 1, 494, 12746, 54746, 31238, 1, 1004, 43628, 330068, 518324, 1, 2026, 143244, 1756878, 5300418, 2620708, 1, 4072, 457536, 8641800, 43235304, 55717312, 1, 8166, 1434318, 40298572, 309074508, 728888188, 325024572

Offset: 0

Alois P. Heinz, Nov 01 2018

			Triangle T(n,k) begins:
  1;
  1;
  1;
  1,     2;
  1,     8;
  1,    22,      22;
  1,    52,     172;
  1,   114,     856,       604;
  1,   240,    3488,      7296;
  1,   494,   12746,     54746,      31238;
  1,  1004,   43628,    330068,     518324;
  1,  2026,  143244,   1756878,    5300418,    2620708;
  1,  4072,  457536,   8641800,   43235304,   55717312;
  1,  8166, 1434318,  40298572,  309074508,  728888188,  325024572;
  1, 16356, 4438540, 180969752, 2026885824, 7589067592, 8460090160;
  ...

Alois P. Heinz, Rows n = 0..100, flattened
S. Spiro, Ballot Permutations, Odd Order Permutations, and a New Permutation Statistic, arXiv preprint arXiv:1810.00993 [math.CO], 2018.
David G. L. Wang, T. Zhao, The peak and descent statistics over ballot permutations, arXiv:2009.05973 [math.CO], 2020.

Columns k=0-3 give: A000012, A005803 (for n>0), A321268, A321269.
Row sums give A000246.
T(2n+1,n) gives A177042.
T(2n+2,n) gives A303285(n+1).
Cf. A262124, A262125.

b:= proc(u, o, c) option remember; `if`(c<0, 0, `if`(u+o=0, 1/x,
       add(expand(x*b(u-j, o-1+j, c-1)), j=1..u)+
       add(b(u+j-1, o-j, c+1), j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(`if`(n=0, 1, b(n, 0, 1))):
seq(T(n), n=0..14);

b[u_, o_, c_] := b[u, o, c] = If[c < 0, 0, If[u + o == 0, 1/x, Sum[Expand[ x*b[u - j, o - 1 + j, c - 1]], {j, 1, u}] + Sum[b[u + j - 1, o - j, c + 1], {j, 1, o}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ If[n == 0, 1, b[n, 0, 1]]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 08 2018, after Alois P. Heinz *)

A321268 Number of permutations on [n] whose up-down signature has nonnegative partial sums and which have exactly two descents.

Original entry on oeis.org

0, 0, 0, 0, 22, 172, 856, 3488, 12746, 43628, 143244, 457536, 1434318, 4438540, 13611136, 41473216, 125797010, 380341580, 1147318004, 3455325600, 10394291094, 31242645420, 93853769320, 281825553760, 846030314842, 2539248578732, 7620161662556, 22865518160768

Offset: 1

Sam Spiro, Nov 01 2018

Also the number of permutations of [n] of odd order whose M statistic (as defined in the Spiro paper) is equal to two.

			Some permutations counted by a(5) include 14253 and 34521.

Sam Spiro, Table of n, a(n) for n = 1..100
S. Spiro, Ballot Permutations, Odd Order Permutations, and a New Permutation Statistic, arXiv:1810.00993 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (11,-50,122,-173,143,-64,12).

Cf. A000246, A005803, A321269, A303285, A177042.

Column k=2 of A321280.

a[1] = 0; a[n_] := 2n^2 - 2n - 1 - n 2^(n-1) - 2 Binomial[n, 3] + Sum[ Binomial[n, k] (2^k - 2k), {k, 0, n}];
Table[a[n], {n, 1, 28}] (* Jean-François Alcover, Nov 11 2018 *)

a(n)={if(n<2, 0, 2*n^2 - 2*n - 1 - n*2^(n-1) - 2*binomial(n,3) + sum(k=0, n, binomial(n, k)*(2^k - 2*k)))} \\ Andrew Howroyd, Nov 01 2018

concat([0,0,0,0], Vec(2*x^5*(11 - 35*x + 32*x^2 - 6*x^3) / ((1 - x)^4*(1 - 2*x)^2*(1 - 3*x)) + O(x^40))) \\ Colin Barker, Mar 07 2019

a(n) = 3*A008292(n-1,3)- 2*binomial(n,3)+binomial(n,2)-1 for n > 1.
a(n) =   A065826(n-1,3)- 2*binomial(n,3)+binomial(n,2)-1 for n > 1.
a(n) = 3^n-3*n*2^(n-1)-2*binomial(n,3)+4*binomial(n,2)-1 for n > 1.
From Colin Barker, Mar 07 2019: (Start)
G.f.: 2*x^5*(11 - 35*x + 32*x^2 - 6*x^3) / ((1 - x)^4*(1 - 2*x)^2*(1 - 3*x)).
a(n) = 11*a(n-1) - 50*a(n-2) + 122*a(n-3) - 173*a(n-4) + 143*a(n-5) - 64*a(n-6) + 12*a(n-7) for n>8.
a(n) = -1 + 3^n - (16+9*2^n)*n/6 + 3*n^2 - n^3/3 for n>1.
(End)

A321269 Number of permutations on [n] whose up-down signature has nonnegative partial sums and which have exactly three descents.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 604, 7296, 54746, 330068, 1756878, 8641800, 40298572, 180969752, 790697160, 3385019968, 14270283414, 59457742524, 245507935018, 1006678811272, 4105447763032, 16672235476128, 67482738851220, 272439143364672, 1097660274098482, 4415486996246052

Offset: 1

Sam Spiro, Nov 01 2018

Also the number of permutations of [n] of odd order whose M statistic (as defined in the Spiro paper) is equal to three.

			The permutations counted by a(7) include 1237654 and 17265243.

Alois P. Heinz, Table of n, a(n) for n = 1..1660
S. Spiro, Ballot Permutations, Odd Order Permutations, and a New Permutation Statistic, arXiv preprint arXiv:1810.00993 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (24,-260,1684,-7278,22172,-49004,79596,-95065,82508,-50616,20800,-5136,576).

Cf. A000246, A005803, A321268, A303285, A177042.

Column k=3 of A321280.

t[n_, k_] := Sum[(-1)^j (k - j)^n Binomial[n + 1, j], {j, 0, k}];
a[n_] := If[n<7, 0, 4 t[n-1, 4] - (Binomial[n, 3] - Binomial[n, 2] + 4) * 2^(n-2) - 22 Binomial[n, 5] + 16 Binomial[n, 4] - 4 Binomial[n, 3] + 2n];
Array[a, 30] (* Jean-François Alcover, Feb 29 2020, from Sam Spiro's 1st formula *)

concat([0,0,0,0,0,0], Vec(2*x^7*(302 - 3600*x + 18341*x^2 - 52006*x^3 + 89327*x^4 - 94728*x^5 + 61016*x^6 - 23368*x^7 + 5424*x^8 - 576*x^9) / ((1 - x)^6*(1 - 2*x)^4*(1 - 3*x)^2*(1 - 4*x)) + O(x^30))) \\ Colin Barker, Mar 07 2019

From Sam Spiro, Mar 07 2019: (Start)
a(n) = 4*A008292(n-1,4)-(binomial(n,3)-binomial(n,2)+4)*2^(n-2)-22*binomial(n,5)+16*binomial(n,4)-4*binomial(n,3)+2n for n>3.
a(n) = A065826(n-1,4)-(binomial(n,3)-binomial(n,2)+4)*2^(n-2)-22*binomial(n,5)+16*binomial(n,4)-4*binomial(n,3)+2n for n>3.
a(n) = 4^n-4*n*3^(n-1)+9*binomial(n,2)*2^(n-2)-binomial(n,3)*2^(n-2)-2^n-8*binomial(n,3)-22*binomial(n,5)+16*binomial(n,4)+2*n for n>3.
(End)
From Colin Barker, Mar 07 2019: (Start)
G.f.: 2*x^7*(302 - 3600*x + 18341*x^2 - 52006*x^3 + 89327*x^4 - 94728*x^5 + 61016*x^6 - 23368*x^7 + 5424*x^8 - 576*x^9) / ((1 - x)^6*(1 - 2*x)^4*(1 - 3*x)^2*(1 - 4*x)).
a(n) = 24*a(n-1) - 260*a(n-2) + 1684*a(n-3) - 7278*a(n-4) + 22172*a(n-5) - 49004*a(n-6) + 79596*a(n-7) - 95065*a(n-8) + 82508*a(n-9) - 50616*a(n-10) + 20800*a(n-11) - 5136*a(n-12) + 576*a(n-13) for n>16.
(End)

More terms from Alois P. Heinz, Nov 01 2018

A181088 a(n) = A181089(2*n+1,n)/(n+2).

Original entry on oeis.org

1, -4, -40, 672, 8064, -253440, -3294720, 153753600, 2091048960, -130025226240, -1820353167360, 141707492720640, 2024392753152000, -189483161695027200, -2747505844577894400, 300609462994993152000, 4408938790593232896000

Offset: 0

Roger L. Bagula, Oct 02 2010

sign

What are the constraints on left-right symmetric triangles t(n,m) such that t(2*n,n)/(n+1) are integers?

G. C. Greubel, Table of n, a(n) for n = 0..450

Cf. A060821, A177042, A177043, A181089.

(* First program *)
p[x_, n_] = HermiteH[n, x] + ExpandAll[x^n*HermiteH[n, 1/x]];
b:= Table[CoefficientList[p[x, n], x], {n, 0, 50}];
Table[b[[2*n+2, n+1]]/(n+2), {n,0,20}]
(* Second program *)
A060821[n_, k_]:= If[EvenQ[n-k], (-1)^(Floor[(n-k)/2])*2^k*n!/(k!*(Floor[(n - k)/2]!)), 0];
a[n_]:= (A060821[2*n+1, n] + A060821[2*n+1, n+1])/(n+2);
Table[a[n], {n, 0, 25}] (* G. C. Greubel, Apr 04 2021 *)

def A060821(n,k): return (-1)^((n-k)//2)*2^k*factorial(n)/(factorial(k)*factorial( (n-k)//2)) if (n-k)%2==0 else 0
def a(n): return (A060821(2*n+1, n) + A060821(2*n+1, n+1))/(n+2)
[a(n) for n in (0..25)] # G. C. Greubel, Apr 04 2021

a(n) = (A060821(2*n+1, n) + A060821(2*n+1, n+1))/(n+2). - G. C. Greubel, Apr 04 2021

A141693 Triangle read by rows: T(n,k) = (2k - n)A008292(n,k) with T(n,n) = n, 0 <= k <= n, where A008292 is the triangle of Eulerian numbers.

Original entry on oeis.org

0, -1, 1, -2, 0, 2, -3, -4, 1, 3, -4, -22, 0, 2, 4, -5, -78, -66, 26, 3, 5, -6, -228, -604, 0, 114, 4, 6, -7, -600, -3573, -2416, 1191, 360, 5, 7, -8, -1482, -17172, -31238, 0, 8586, 988, 6, 8, -9, -3514, -73040, -264702, -156190, 88234, 43824, 2510, 7, 9, -10

Offset: 0

Roger L. Bagula, Sep 09 2008

			Triangle begins:
    0;
   -1,     1;
   -2,     0,      2;
   -3,    -4,      1,       3;
   -4,   -22,      0,       2,       4;
   -5,   -78,    -66,      26,       3,     5;
   -6,  -228,   -604,       0,     114,     4,    6;
   -7,  -600,  -3573,   -2416,    1191,   360,    5,     7;
   -8, -1482, -17172,  -31238,       0,  8586,  988,     6, 8;
   -9, -3514, -73040, -264702, -156190, 88234, 43824, 2510, 7, 9;
  ...

Eric Weisstein's World of Mathematics, Eulerian Number
Wikipedia, Eulerian number

Cf. A008292.

T:= proc(n,k) `if`(n=k,n,(2*k-n)*add((-1)^j*(k-j+1)^n*binomial(n+1,j),j=0..k)); end proc: seq(seq(T(n,k),k=0..n),n=0..10); # Muniru A Asiru, Oct 06 2018
T := (n, k) -> `if`(n = k, n, (2*k - n)*combinat:-eulerian1(n,k)):
seq(seq(T(n,k), k=0..n), n=0..9); # Peter Luschny, Oct 06 2018

T[n_, k_] = If[n == k, n, (2*k - n)*Sum[(-1)^j*(k - j + 1)^n*Binomial[n + 1, j], {j, 0, k}]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}]//Flatten

T(n, k) := if n = k then n else (2*k - n)*sum((-1)^j*(k - j + 1)^n*binomial(n + 1, j), j, 0, k)$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$ /* Franck Maminirina Ramaharo, Oct 05 2018 */

Sum_{k=0..n} T(n,k) = A005096(n), n > 0.
From Franck Maminirina Ramaharo, Oct 06 2018: (Start)
T(n,k) = (2*k - n)*Sum_{j=0..k} (-1)^j*(k - j + 1)^n*binomial(n + 1, j) for 0 <= k <= n - 1 and T(n,n) = n.
T(2*n-1,n-1) = -A025585(n).
T(2*n,n-1) = -A177042(n). (End)

Edited, new name and offset corrected by Franck Maminirina Ramaharo, Oct 06 2018

cp's OEIS Frontend

A008518 Triangle of Eulerian numbers with rows multiplied by 1 + x.

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A303285 Number of permutations p of [2n] such that the sequence of ascents and descents of p0 forms a Dyck path.

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A303287 Number of permutations p of [n] such that the sequence of ascents and descents of p or of p0 (if n is even) forms a Dyck path.

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A316728 Number T(n,k) of permutations of {0,1,...,2n} with first element k whose sequence of ascents and descents forms a Dyck path; triangle T(n,k), n>=0, 0<=k<=2n, read by rows.

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A172010 a(n) = 2*A142458(2*n, n)/(n+1).

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A321280 Number T(n,k) of permutations p of [n] with exactly k descents such that the up-down signature of p has nonnegative partial sums; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/2)), read by rows.

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A321268 Number of permutations on [n] whose up-down signature has nonnegative partial sums and which have exactly two descents.

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A172010 a(n) = 2A142458(2n, n)/(n+1).

A141693 Triangle read by rows: T(n,k) = (2k - n)A008292(n,k) with T(n,n) = n, 0 <= k <= n, where A008292 is the triangle of Eulerian numbers.