A001793 -id:A001793 - cp's OEIS frontend

A089741 Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n containing k UHH...HD's, where U=(1,1), D=(1,-1) and H=(1,0) (can be easily expressed using RNA secondary structure terminology).

1, 1, 1, 1, 1, 1, 3, 1, 7, 1, 15, 1, 1, 31, 5, 1, 63, 18, 1, 127, 56, 1, 1, 255, 160, 7, 1, 511, 432, 34, 1, 1023, 1120, 138, 1, 1, 2047, 2816, 500, 9, 1, 4095, 6912, 1672, 55, 1, 8191, 16640, 5264, 275, 1, 1, 16383, 39424, 15808, 1205, 11, 1, 32767, 92160, 45696, 4797

Offset: 0

Emeric Deutsch, Jan 08 2004

			T(7,2)=5 because we have H(UHD)(UHD), (UHD)H(UHD), (UHD)(UHD)H, (UHD)(UHHD) and (UHHD)(UHD) (the required subwords are shown between parentheses).
Triangle begins:
  1;
  1;
  1;
  1,   1;
  1,   3;
  1,   7;
  1,  15,   1;
  1,  31,   5;
  1,  63,  18;
  1, 127,  56,   1;
  1, 255, 160,   7;
  ...

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.
M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08; Sem. Loth. Comb. B08l (1984) 79-86.

Row sums yield A004148. Column 1 is A000225, column 2 is A001793.

G.f.: (1-2*z+2*z^2-t*z^3-sqrt((1-t*z^3)*(1-4*z+4*z^2-t*z^3)))/(2*z^2*(1-z)).

A097229 Triangle read by rows: number of Motzkin paths by length and by number of humps.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 7, 1, 1, 15, 5, 1, 31, 18, 1, 1, 63, 56, 7, 1, 127, 160, 34, 1, 1, 255, 432, 138, 9, 1, 511, 1120, 500, 55, 1, 1, 1023, 2816, 1672, 275, 11, 1, 2047, 6912, 5264, 1205, 81, 1, 1, 4095, 16640, 15808, 4797, 481, 13

Offset: 0

David Callan, Aug 01 2004

T(n,k) = number of Motzkin paths of length n containing exactly k humps. (A hump is an upstep followed by 0 or more flatsteps followed by a downstep.)

			Example: Table begins
  n|
  -+------------------
  0|1
  1|1
  2|1,   1
  3|1,   3
  4|1,   7,   1
  5|1,  15,   5
  6|1,  31,  18,  1
  7|1,  63,  56,  7
  8|1, 127, 160, 34, 1
T(5,2) = 5 counts FUDUD, UDFUD, UDUDF, UDUFD, UFDUD.

Yan Zhuang, A generalized Goulden-Jackson cluster method and lattice path enumeration, Discrete Mathematics 341.2 (2018): 358-379; arXiv:1508.02793 [math.CO], 2015-2018.

Column k=2 is A001793.

Cf. A001263.

a[n_, k_]/;k<0 || k>n/2 := 0; a[n_, 0]/;n>=0 := 1; a[n_, k_]/;1<=k<=n := a[n, k] = a[n-1, k] + Sum[a[n-r, k-1], {r, 2, n}]+Sum[a[r-2, j]a[n-r, k-j], {r, 2, n}, {j, k}] (* This recurrence counts a(n, k) by first return to ground level. *)

N(n,k):=(binomial(n,k-1)*binomial(n,k))/n;
T(n,k):=if k=0 then 1 else sum(binomial(n, 2*i)*N(i,k),i,1,n); /* Vladimir Kruchinin, Jan 08 2022 */

G.f.: ((-1 + 2*x - 2*x^2 + x^2*y + ((1 - 2*x)^2 + 2*x^2*(-1 + 2*x - 2*x^2)*y + x^4*y^2)^(1/2))/(2*(-1 + x)*x^2) = Sum_{n>=0, k>=0} a(n, k) x^n y^k satisfies x^2 A(x, y)^2 - ( x^2(1-y)/(1-x) + (1-x) )A(x, y) + 1 = 0.

T(n,k) = Sum_{i=1..n} binomial(n, 2*i)*N(i,k), T(n,0)=1, where N(n,k) is the triangle of Narayana numbers A001263. - Vladimir Kruchinin, Jan 08 2022

A102841 a(n) = ((9n^2 + 33n + 26)*2^n + (-1)^n)/27.

Original entry on oeis.org

1, 5, 19, 61, 179, 493, 1299, 3309, 8211, 19949, 47635, 112109, 260627, 599533, 1366547, 3089901, 6937107, 15476205, 34331155, 75769325, 166451731, 364127725, 793500179, 1723082221, 3729512979, 8048092653, 17319057939

Offset: 0

Creighton Dement, Feb 27 2005

nonn

A floretion-generated sequence relating the number of edges and faces in n-dimensional hypercube.
Equals A001787, (1, 4, 12, 32, 80, ...) convolved with A001045, the Jacobsthal sequence. - Gary W. Adamson, May 23 2009
The sum of the sizes of all inversions in compositions of n. - Arnold Knopfmacher, Jan 22 2020

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. Archibald, A. Blecher, A. Knopfmacher, M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
Index entries for linear recurrences with constant coefficients, signature (5,-6,-4,8).

Cf. A045883, A001788, A001793, A102301.

Cf. A001787, A001045. - Gary W. Adamson, May 23 2009

[((9*n^2 + 33*n + 26)*2^n + (-1)^n)/27 : n in [0..40]]; // Wesley Ivan Hurt, Jul 03 2020

Table[(1/27)*((9 n^2 + 33 n + 26) 2^n + (-1)^n), {n, 0, 50}] (* or *) LinearRecurrence[{5,-6,-4,8}, {1,5,19,61}, 50] (* G. C. Greubel, Sep 27 2017 *)

G.f.: 1/((1+x)*(1-2*x)^3).
a(n+1) - 2*a(n) = A045883(n+2).
a(n) + a(n+1) = A001788(n+2).
a(n) = 5*a(n-1) - 6*a(n-2) - 4*a(n-3) + 8*a(n-4). - Wesley Ivan Hurt, Jul 03 2020

Corrected by T. D. Noe, Nov 08 2006

A114593 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n, having k ascents of length at least 2 (1 <= k <= floor(n/2), n >= 2).

Original entry on oeis.org

1, 2, 4, 2, 8, 10, 16, 36, 5, 32, 112, 42, 64, 320, 224, 14, 128, 864, 960, 168, 256, 2240, 3600, 1200, 42, 512, 5632, 12320, 6600, 660, 1024, 13824, 39424, 30800, 5940, 132, 2048, 33280, 119808, 128128, 40040, 2574, 4096, 78848, 349440, 489216, 224224, 28028, 429

Offset: 2

Emeric Deutsch, Dec 11 2005

Row n has floor(n/2) terms. Row sums are the Fine numbers (A000957). T(n,1) = 2^(n-2). T(n,2) = n(n-3)2^(n-5) (n>4) (2*A001793). T(2n,n) = Catalan(n). T(2n+1,n) = n*Catalan(n+1). Sum_{k=1..floor(n/2)} k*T(n,k) yields A114594.

T(n,k) is the number of permutations pi of [n-1] with k valleys such that s(pi) avoids the patterns 132, 231, and 312, where s is West's stack-sorting map. - Colin Defant, Sep 16 2018

			T(4,2)=2 because we have (UU)D(UU)DDD and (UU)DD(UU)DD, where U=(1,1), D=(1,-1) (ascents of length at least two are shown between parentheses).
Triangle starts:
   1;
   2;
   4,   2;
   8,  10;
  16,  36,   5;
  32, 112,  42;
  64, 320, 224,  14;

Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.

Cf. A000957, A001793, A114594.

T:=proc(n,k) if k<=floor(n/2) then 2^(n-2*k)*binomial(n+1,k)*binomial(n-k-1,k-1)/(n+1) else 0 fi end: for n from 2 to 14 do seq(T(n,k),k=1..floor(n/2)) od;

m = 13(*rows*); G = 0; Do[G = Series[(1 + G^2 (2 + t z) z)/(1 + 2 z), {t, 0, m+1}, {z, 0, m+1}] // Normal // Expand, {m+2}]; Rest[CoefficientList[ #, t]]& /@ CoefficientList[G-1, z][[3;;]] // Flatten (* Jean-François Alcover, Jan 22 2019 *)

T(n, k) = 2^(n-2k)*binomial(n+1, k)*binomial(n-k-1, k-1)/(n+1) (1 <= k <= floor(n/2)). G.f. = G-1, where G=G(t, z) satisfies z(2+tz)G^2 - (1+2z)G + 1 = 0.

A136523 Triangle T(n,k) = A053120(n,k) + A053120(n-1,k), read by rows.

Original entry on oeis.org

1, 1, 1, -1, 1, 2, -1, -3, 2, 4, 1, -3, -8, 4, 8, 1, 5, -8, -20, 8, 16, -1, 5, 18, -20, -48, 16, 32, -1, -7, 18, 56, -48, -112, 32, 64, 1, -7, -32, 56, 160, -112, -256, 64, 128, 1, 9, -32, -120, 160, 432, -256, -576, 128, 256, -1, 9, 50, -120, -400, 432, 1120, -576, -1280, 256, 512

Offset: 0

Roger L. Bagula, Mar 23 2008

			Triangle begins as:
   1;
   1,  1;
  -1,  1,   2;
  -1, -3,   2,    4;
   1, -3,  -8,    4,    8;
   1,  5,  -8,  -20,    8,   16;
  -1,  5,  18,  -20,  -48,   16,   32;
  -1, -7,  18,   56,  -48, -112,   32,   64;
   1, -7, -32,   56,  160, -112, -256,   64,   128;
   1,  9, -32, -120,  160,  432, -256, -576,   128, 256;
  -1,  9,  50, -120, -400,  432, 1120, -576, -1280, 256, 512;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000007, A001792, A001793, A001794, A000330, A008794, A011782.
Cf. A025192, A040000 (row sums), A053120, A057077, A081277, A109613.
Cf. A124182.

function A053120(n,k)
  if ((n+k) mod 2) eq 1 then return 0;
  elif n eq 0 then return 1;
  else return (-1)^Floor((n-k)/2)*(n/(n+k))*Binomial(Floor((n+k)/2), k)*2^k;
  end if;
end function;
A136523:= func< n,k | A053120(n,k) + A053120(n-1,k) >;
[A136523(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 26 2023

A053120[n_, k_]:= Coefficient[ChebyshevT[n,x], x, k];
T[n_, k_]:= T[n, k]= A053120[n,k] + A053120[n-1,k];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

def A053120(n,k):
    if (n+k)%2==1: return 0
    elif n==0: return 1
    else: return floor((-1)^((n-k)//2)*(n/(n+k))*binomial((n+k)//2, k)*2^k)
def A136523(n,k): return A053120(n,k) + A053120(n-1,k)
flatten([[A136523(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 26 2023

T(n, k) = A053120(n,k) + A053120(n-1,k).
Sum_{k=0..n} T(n, k) = A040000(n).
From G. C. Greubel, Jul 26 2023: (Start)
T(n, 0) = A057077(n).
T(n, 1) = (-1)^floor((n-1)/2) * A109613(n-1).
T(n, 2) = (-1)^floor((n-2)/2) * A008794(n-1).
T(n, 3) = (-1)^floor((n+1)/2) * A000330(n-1).
T(n, n) = A011782(n).
T(n, n-1) = A011782(n-1).
T(n, n-2) = -A001792(n-2).
T(n, n-4) = A001793(n-3).
T(n, n-6) = -A001794(n-6).
Sum_{k=0..n} (-1)^k*T(n,k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000007(n) + [n=1].
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^floor(n/2)*A025192(floor(n/2)). (End)

Edited by G. C. Greubel, Jul 26 2023

A224288 Number of permutations of length n containing exactly 2 occurrences of 123 and 2 occurrences of 132.

Original entry on oeis.org

0, 0, 0, 0, 1, 6, 26, 94, 306, 934, 2732, 7752, 21488, 58432, 156288, 411904, 1071104, 2750976, 6984704, 17545216, 43634688, 107511808, 262602752, 636223488, 1529741312, 3652059136, 8660975616, 20412104704, 47826599936, 111446851584, 258360737792, 596044152832

Offset: 0

Brian Nakamura, Apr 03 2013

			a(4) = 1: (1,2,4,3).
a(5) = 6: (2,3,5,1,4), (2,3,5,4,1), (2,5,1,3,4), (3,1,4,5,2), (4,1,2,5,3), (5,1,2,4,3).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
B. Nakamura, Approaches for enumerating permutations with a prescribed number of occurrences of patterns, arXiv 1301.5080 [math.CO], 2013.
B. Nakamura, A Maple package for enumerating n-permutations with r occurrences of the pattern 123 and s occurrences of the pattern 132 [Broken link]
Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

Cf. A000079, A001787, A001815, A046718, A001793.

# Programs can be obtained from the Nakamura link

Join[{0, 0, 0, 0, 1}, LinearRecurrence[{10, -40, 80, -80, 32}, {6, 26, 94, 306, 934}, 27]] (* Jean-François Alcover, Feb 29 2020 *)

G.f.: -(2*x^5+6*x^4-6*x^3+6*x^2-4*x+1)*x^4/(2*x-1)^5. - Alois P. Heinz, Apr 03 2013

a(n) = 2^(-11+n)*(1504-994*n+219*n^2-18*n^3+n^4) for n>4. - Colin Barker, Apr 14 2013

A224290 Number of permutations of length n containing exactly 3 occurrences of 123 and 3 occurrences of 132.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 6, 30, 136, 566, 2176, 7808, 26440, 85332, 264632, 793792, 2315136, 6592640, 18390784, 50392064, 135921664, 361536512, 949708800, 2466807808, 6342115328, 16153509888, 40790523904, 102186352640, 254105092096, 627533152256, 1539764125696

Offset: 0

Brian Nakamura, Apr 03 2013

nonn

			a(5) = 1: (1,4,3,2,5).
a(6) = 6: (2,5,4,3,1,6), (2,5,4,3,6,1), (3,5,1,4,6,2), (3,6,1,4,2,5), (5,1,4,3,2,6), (6,1,4,3,2,5).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
B. Nakamura, Approaches for enumerating permutations with a prescribed number of occurrences of patterns, arXiv 1301.5080 [math.CO], 2013.
B. Nakamura, A Maple package for enumerating n-permutations with r occurrences of the pattern 123 and s occurrences of the pattern 132 [Broken link]
Index entries for linear recurrences with constant coefficients, signature (14,-84,280,-560,672,-448,128).

Cf. A000079, A001787, A001815, A046718, A001793.

# Programs can be obtained from the Nakamura link

Join[{0, 0, 0, 0, 0, 1, 6}, LinearRecurrence[{14, -84, 280, -560, 672, -448, 128}, {30, 136, 566, 2176, 7808, 26440, 85332}, 33]] (* Jean-François Alcover, Nov 28 2018 *)

concat([0,0,0,0,0], Vec(x^5*(1 - 8*x + 30*x^2 - 60*x^3 + 62*x^4 - 36*x^5 + 24*x^6 - 8*x^7 + 4*x^8) / (1 - 2*x)^7 + O(x^40))) \\ Colin Barker, Nov 28 2018

G.f.: -(4*x^8-8*x^7+24*x^6-36*x^5+62*x^4-60*x^3+30*x^2-8*x+1)*x^5 / (2*x-1)^7. - Alois P. Heinz, Apr 03 2013
From Colin Barker, Nov 28 2018: (Start)
a(n) = (1/9)*2^(n-15) * (307008 - 247512*n + 78118*n^2 - 12087*n^3 + 937*n^4 - 33*n^5 + n^6) for n>6.
a(n) = 14*a(n-1) - 84*a(n-2) + 280*a(n-3) - 560*a(n-4) + 672*a(n-5) - 448*a(n-6) + 128*a(n-7) for n>13.
(End)

A304635 Triangle T(n,j) read by rows: the number of j-faces in the hypersimplicial decomposition of the unit cube of n dimensions.

Original entry on oeis.org

1, 5, 2, 18, 14, 3, 56, 64, 27, 4, 160, 240, 150, 44, 5, 432, 800, 660, 288, 65, 6, 1120, 2464, 2520, 1456, 490, 90, 7, 2816, 7168, 8736, 6272, 2800, 768, 119, 8, 6912, 19968, 28224, 24192, 13440, 4896, 1134, 152, 9, 16640, 53760, 86400, 86016, 57120, 25920, 7980, 1600, 189, 10

Offset: 1

R. J. Mathar, May 15 2018

			The triangle starts in row n>= for 1<=j<=n as:
  1,
5,2,
18,14,3,
56,64,27,4,
160,240,150,44,5,
432,800,660,288,65,6,
1120,2464,2520,1456,490,90,7,
2816,7168,8736,6272,2800,768,119,8,
6912,19968,28224,24192,13440,4896,1134,152,9,
16640,53760,86400,86016,57120,25920,7980,1600,189,10,

T. Hibi, N. Li, H. Ohsugi, The Face Vector of a Half-Open Hypersimplex, J. Int. Seq. 18 (2015) 15.6.6

Cf. A001793 (column j=1), A001794 (half of column j=2), A006974 (3rd of column j=3), A014106 (subdiagonal).

A304635 := proc(n,j)
        j*2^(n-j-1)*(n+j+2)/(n+1)*binomial(n+1,j+1) ;
end proc:

T(n,j) = j*2^(n-j-1)*(n+j+2)*binomial(n+,j+1)/(n+1).

A349427 a(n) = ((n+1)^2 - 2) * binomial(2*n-2,n-1) / 2.

Original entry on oeis.org

0, 1, 7, 42, 230, 1190, 5922, 28644, 135564, 630630, 2892890, 13117676, 58903572, 262303132, 1159666900, 5094808200, 22259364120, 96773942790, 418882316490, 1805951924700, 7758285404100, 33221013445620, 141830949914940, 603876402587640, 2564713671647400

Offset: 0

Ilya Gutkovskiy, Nov 17 2021

Cf. A000108, A000217, A000984, A001700, A001793, A002457, A002544, A008865, A037966, A088218, A127736.

Table[((n + 1)^2 - 2) Binomial[2 n - 2, n - 1]/2, {n, 0, 24}]
nmax = 24; CoefficientList[Series[x (1 - x) (1 - 2 x)/(1 - 4 x)^(5/2), {x, 0, nmax}], x]

a(n) = ((n+1)^2 - 2) * binomial(2*n-2,n-1) / 2 \\ Andrew Howroyd, Nov 20 2021

G.f.: x * (1 - x) * (1 - 2*x) / (1 - 4*x)^(5/2).
E.g.f.: x * exp(2*x) * (2 * (1 + x) * BesselI(0,2*x) + (1 + 2*x) * BesselI(1,2*x)) / 2.
a(n) = n * ((n+1)^2 - 2) * Catalan(n-1) / 2.
a(n) = Sum_{k=0..n} binomial(n,k)^2 * A000217(k).
a(n) ~ 2^(2*n-3) * n^(3/2) / sqrt(Pi).
D-finite with recurrence (n-1)*(n^2-2)*a(n) -2*(2*n-3)*(n^2+2*n-1)*a(n-1)=0. - R. J. Mathar, Mar 06 2022

A091977 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k exterior pairs.

Original entry on oeis.org

1, 1, 2, 4, 1, 8, 5, 1, 16, 18, 7, 1, 32, 56, 34, 9, 1, 64, 160, 138, 55, 11, 1, 128, 432, 500, 275, 81, 13, 1, 256, 1120, 1672, 1205, 481, 112, 15, 1, 512, 2816, 5264, 4797, 2471, 770, 148, 17, 1, 1024, 6912, 15808, 17738, 11403, 4536, 1156, 189, 19, 1, 2048, 16640

Offset: 0

Emeric Deutsch, Mar 15 2004

A pyramid in a Dyck word (path) is a factor of the form u^h d^h, h being the height of the pyramid. A pyramid in a Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a u and immediately followed by a d.
The pyramid weight of a Dyck path (word) is the sum of the heights of its maximal pyramids. An exterior pair in a Dyck path is a pair consisting of a u and its matching d (when viewed as parentheses) which do not belong in any pyramid. Clearly, for a given Dyck path, the sum of its pyramid weight and the number of its exterior pairs is equal to the semilength of the path.
Triangle, with zeros omitted, given by (1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, ...) DELTA (0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 06 2012

			T(4,1)=5 because the Dyck paths of semilength 4 having 1 exterior pair are: ud(u)udud(d), (u)udud(d)ud, (u)ududud(d), (u)uduudd(d) and (u)uuuddud(d) [the u and d that form the unique exterior pair are shown between parentheses].
Triangle begins:
[1],
[1],
[2],
[4, 1],
[8, 5, 1],
[16, 18, 7, 1],
[32, 56, 34, 9, 1],
[64, 160, 138, 55, 11, 1],
[128, 432, 500, 275, 81, 13, 1]
Triangle (1,1,0,1,1,0,1,1,...) DELTA (0,0,1,0,0,1,0,0,1,...) begins :
1
1, 0
2, 0, 0
4, 1, 0, 0
8, 5, 1, 0, 0
16, 18, 7, 1, 0, 0
32, 56, 34, 9, 1, 0, 0
64, 160, 138, 55, 11, 1, 0, 0...- _Philippe Deléham_, Feb 06 2012

A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.

T(n, k)=A091866(n, n-k), T(n, 0)=2^(n-1) (n>0), T(n, 1)=A001793(n-2), row sums give the Catalan numbers (A000108).

Cf. A091866, A001793, A000108.

G.f.=G=G(t, z) satisfies tz(1-z)G^2-(1+tz-2z)G+1-z=0.

cp's OEIS Frontend

A089741 Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n containing k UHH...HD's, where U=(1,1), D=(1,-1) and H=(1,0) (can be easily expressed using RNA secondary structure terminology).

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A097229 Triangle read by rows: number of Motzkin paths by length and by number of humps.

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Mathematica

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A102841 a(n) = ((9*n^2 + 33*n + 26)*2^n + (-1)^n)/27.

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A114593 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n, having k ascents of length at least 2 (1 <= k <= floor(n/2), n >= 2).

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A136523 Triangle T(n,k) = A053120(n,k) + A053120(n-1,k), read by rows.

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SageMath

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A224288 Number of permutations of length n containing exactly 2 occurrences of 123 and 2 occurrences of 132.

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A224290 Number of permutations of length n containing exactly 3 occurrences of 123 and 3 occurrences of 132.

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PARI

A102841 a(n) = ((9n^2 + 33n + 26)*2^n + (-1)^n)/27.