A001628 -id:A001628 - cp's OEIS frontend

A261054 Expansion of ( 2+x-x^2+x^3 ) / (1-x^2-x)^3 .

2, 7, 20, 51, 118, 260, 548, 1121, 2236, 4373, 8412, 15960, 29926, 55547, 102196, 186567, 338258, 609532, 1092328, 1947829, 3457724, 6112873, 10766328, 18896880, 33062090, 57675631, 100338836, 174117915, 301432558, 520686644, 897559340, 1544207369, 2651881468, 4546263293, 7781245524

Offset: 0

R. J. Mathar, Aug 08 2015

Third column of A234713.

Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Cf. A234713.

LinearRecurrence[{3,0,-5,0,3,1},{2,7,20,51,118,260},40] (* Harvey P. Dale, Aug 10 2024 *)

a(n) = 2*A001628(n) +A001628(n-1) -A001628(n-2) +A001628(n-3).

A261055 Expansion of ( -1-2*x+x^2+x^3 ) / (x^2+x-1)^3 .

Original entry on oeis.org

1, 5, 14, 36, 83, 182, 382, 778, 1546, 3013, 5778, 10932, 20447, 37867, 69526, 126690, 229309, 412570, 738308, 1314824, 2331218, 4116731, 7243092, 12700680, 22201165, 38696513, 67267550, 116642832, 201791063, 348339614, 600095386, 1031830006

Offset: 0

R. J. Mathar, Aug 08 2015

Third column of A228815.

Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Cf. A228815.

CoefficientList[Series[(-1 - 2 x + x^2 + x^3)/(x^2 + x - 1)^3, {x, 0, 40}], x] (* or *) LinearRecurrence[{3,0,-5,0,3,1},{1,5,14,36,83,182},40] (* Harvey P. Dale, Nov 21 2018 *)

a(n) = A001628(n) +2*A001628(n-1) -A001628(n-2) -A001628(n-3).

A114711 Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n and having k weak ascents (1 <= k <= ceiling(n/3)).

Original entry on oeis.org

1, 1, 2, 3, 1, 5, 3, 8, 9, 13, 22, 2, 21, 51, 10, 34, 111, 40, 55, 233, 130, 5, 89, 474, 380, 35, 144, 942, 1022, 175, 233, 1836, 2590, 700, 14, 377, 3522, 6260, 2450, 126, 610, 6666, 14570, 7770, 756, 987, 12473, 32870, 22890, 3570, 42, 1597, 23109, 72244

Offset: 1

Emeric Deutsch, Dec 27 2005

A Motzkin path of length n is a lattice path from (0,0) to (n,0) consisting of U=(1,1), D=(1,-1) and H=(1,0) steps and never going below the x-axis. A weak ascent in a Motzkin path is a maximal sequence of consecutive U and H steps.

			T(5,2)=3 because we have (UH)D(UU), (UHH)D(H) and (HUH)D(H) (the weak ascents are shown between parentheses).
Triangle begins:
   1;
   1;
   2;
   3,  1;
   5,  3;
   8,  9;
  13, 22,  2;

Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
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 (1978), 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, Actes 8e Sem. Lotharingien, pp. 79-86.

Cf. A004148, A000045, A001628, A000108, A051286.

G:=(1-z-z^2-sqrt(1-2*z-z^2+2*z^3+z^4-4*t*z^3))/2/z^2: Gser:=simplify(series(G,z=0,22)): for n from 1 to 18 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 18 do seq(coeff(P[n],t^j),j=1..ceil(n/3)) od; # yields sequence in triangular form

Row n contains ceiling(n/3) terms.
Row sums yield the RNA secondary structure numbers (A004148).
Column 1 yields the Fibonacci numbers (A000045).
Column 2 yields A001628.
T(3n+1,n+1) = A000108(n) (the Catalan numbers).
Sum_{k=1..ceiling(n/3)} k*T(n,k) = A051286(n-1) (n >= 1).
G.f.: G = G(t, z) satisfies G = z*(t+G) + z^2*G*(1+G).

A132886 Triangle read by rows: T(n,k) is the number of paths in the right half-plane, from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having k U steps (0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 2, 2, 3, 6, 5, 18, 6, 8, 44, 30, 13, 102, 120, 20, 21, 222, 390, 140, 34, 466, 1140, 700, 70, 55, 948, 3066, 2800, 630, 89, 1884, 7770, 9800, 3780, 252, 144, 3672, 18780, 31080, 17850, 2772, 233, 7044, 43710, 91560, 72450, 19404, 924, 377, 13332, 98610

Offset: 0

Emeric Deutsch, Sep 03 2007

Row n has 1+floor(n/2) terms. T(n,0) = A000045(n+1) (the Fibonacci numbers). T(2n,n) = binomial(2n,n) = A000984(n) (the central binomial coefficients). Row sums yield A059345. Column k has g.f. = binomial(2k,k)*z^(2k)/(1-z-z^2)^(2k+1); accordingly, T(n,1) = 2*A001628(n-2), T(n,2) = 6*A001873(n-4), T(n,3) = 20*A001875(n-6). See A132883 for the same statistic on paths restricted to the first quadrant.

			Triangle starts:
   1;
   1;
   2,   2;
   3,   6;
   5,  18,   6;
   8,  44,  30;
  13, 102, 120,  20;
T(3,1)=6 because we have hUD, UhD, UDh, hDU, DhU and DUh.

Cf. A000045, A059345, A001628, A001873, A001875, A132883.

G:=1/sqrt((1-z-z^2)^2-4*t*z^2): Gser:=simplify(series(G,z=0,17)): for n from 0 to 13 do P[n]:= sort(coeff(Gser,z,n)) end do: for n from 0 to 13 do seq(coeff(P[n],t,j),j=0..floor((1/2)*n)) end do; # yields sequence in triangular form

G.f.: G(t,z) = 1/sqrt((1-z-z^2)^2 - 4tz^2).

A181974 Triangle T(n,k), read by rows, given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -3, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 4, 2, 1, 5, 7, 5, 4, 1, 8, 11, 10, 9, 3, 1, 13, 18, 20, 20, 9, 5, 1, 21, 29, 38, 40, 22, 15, 4, 1, 34, 47, 71, 78, 51, 40, 14, 6, 1, 55, 76, 130, 147, 111, 95, 40, 22, 5, 1, 89, 123, 235, 272, 233, 213, 105, 68, 20, 7, 1

Offset: 0

Philippe Deléham, Apr 06 2012

			Triangle begins :
1
1, 1
2, 3, 1
3, 4, 2, 1
5, 7, 5, 4, 1
8, 11, 10, 9, 3, 1
13, 18, 20, 20, 9, 5, 1
21, 29, 38, 40, 22, 15, 4, 1
34, 47, 71, 78, 51, 40, 14, 6, 1
55, 76, 130, 147, 111, 95, 40, 22, 5, 1
89, 123, 235, 272, 233, 213, 105, 68, 20, 7, 1
144, 199, 420, 495, 474, 455, 256, 185, 65, 30, 6, 1

Cf. Columns : A000045, A000032, A001629, A023607, A001628, A152881, A001872, A001873

G.f.: (1+y*x+2*y*x^2)/(1-x-x^2-y^2*x^2).
T(n,k) = T(n-1,k) + T(n-2,k) + T(n-2,k-2), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = 2, T(2,1) = 3 and T(n,k) = 0 if k<0 or if k>n.
T(n + 2k, 2k) = A037027(n + k, k).
T(n + 2k +1, 2k + 1) = A182001(n + k, k).
T(n,0) = Fibonacci(n+1).

A185664 Riordan array (A000045(x)^m,x*A000108(x)), m=3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 0, 9, 4, 1, 0, 0, 0, 22, 14, 5, 1, 0, 0, 0, 51, 42, 20, 6, 1, 0, 0, 0, 111, 120, 69, 27, 7, 1, 0, 0, 0, 233, 335, 224, 104, 35, 8, 1, 0, 0, 0, 474, 940, 707, 372, 148, 44, 9, 1, 0, 0, 0, 942, 2695, 2221, 1281, 574, 202, 54, 10, 1, 0, 0, 0, 1836, 7980, 7038, 4343, 2122, 841, 267, 65, 11, 1, 0, 0, 0

Offset: 0

Vladimir Kruchinin, Feb 08 2011

			0;
0,0;
0,0,0;
1,0,0,0;
3,1,0,0,0;
9,4,1,0,0,0;
22,14,5,1,0,0,0;
51,42,20,6,1,0,0,0;

Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A001628 (column k=0).

R(n,k,m)=k*sum(i=0..n-k, sum(j=ceiling((i-m)/2)..i-m, binomial(j,i-m-j)*binomial(m+j-1,m-1))*binomial(2*(n-i)-k-1,n-i-1)/(n-i)), k>0.

R(n,0,m)=sum(j=ceiling((i-m)/2)..n-m, binomial(j,n-m-j)*binomial(m+j-1,m-1)), m=3.

cp's OEIS Frontend

A261054 Expansion of ( 2+x-x^2+x^3 ) / (1-x^2-x)^3 .

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A261055 Expansion of ( -1-2*x+x^2+x^3 ) / (x^2+x-1)^3 .

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A114711 Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n and having k weak ascents (1 <= k <= ceiling(n/3)).

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A132886 Triangle read by rows: T(n,k) is the number of paths in the right half-plane, from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having k U steps (0 <= k <= floor(n/2)).

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A181974 Triangle T(n,k), read by rows, given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -3, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A185664 Riordan array (A000045(x)^m,x*A000108(x)), m=3.

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