A086615 -id:A086615 - cp's OEIS frontend

A182013 Triangle of partial sums of Motzkin numbers.

1, 2, 1, 4, 3, 2, 8, 7, 6, 4, 17, 16, 15, 13, 9, 38, 37, 36, 34, 30, 21, 89, 88, 87, 85, 81, 72, 51, 216, 215, 214, 212, 208, 199, 178, 127, 539, 538, 537, 535, 531, 522, 501, 450, 323, 1374, 1373, 1372, 1370, 1366, 1357, 1336, 1285, 1158, 835, 3562, 3561

Offset: 0

Emanuele Munarini, Apr 06 2012

			Triangle begins:
  1
  2,   1
  4,   3,   2
  8,   7,   6,   4
  17,  16,  15,  13,  9
  38,  37,  36,  34,  30,  21
  89,  88,  87,  85,  81,  72,  51
  216, 215, 214, 212, 208, 199, 178, 127
  539, 538, 537, 535, 531, 522, 501, 450, 323

Diagonal elements = Motzkin numbers (A001006).
First column = partial sums of Motzkin numbers (A086615).
Row sums = A097861(n+1).
Diagonal sums = A182015.
Row square-sums = A182017.
Central coefficients = A182016.

M[n_] := If[n==0, 1, Coefficient[(1+x+x^2)^(n+1), x^n]/(n+1)]; Flatten[Table[Sum[M[i], {i,k,n}], {n,0,30}, {k,0,n}]]

M(n):=coeff(expand((1+x+x^2)^(n+1)),x^n)/(n+1);
create_list(sum(M(i),i,k,n),n,0,6,k,0,n);

T(n, k) = Sum_{i=k..n} M(i), where the M(n)'s are the Motzkin numbers.
Recurrence: T(n+1, k+1) = T(n, k) + M(n+1) - M(k).
G.f. (M(x) - y*M(x*y))/((1 - x)*(1 - y)), where M(x) is the generating series for Motzkin numbers.

A348869 Triangle T(n,c) counting Motzkin Paths of length n with c sections starting with an up-step at level 0.

Original entry on oeis.org

1, 2, 4, 1, 8, 4, 17, 12, 1, 38, 32, 6, 89, 82, 24, 1, 216, 208, 80, 8, 539, 530, 243, 40, 1, 1374, 1364, 702, 160, 10, 3562, 3551, 1975, 564, 60, 1, 9360, 9348, 5484, 1840, 280, 12, 24871, 24858, 15144, 5716, 1125, 84, 1, 66706, 66692, 41768, 17208, 4102, 448, 14

Offset: 2

R. J. Mathar, Nov 02 2021

This is a Sequence Transform of A086615. A086615(n-2) counts the Motzkin Paths of length n which start with an u-step, return to the horizontal level once with a d-step and remain there (with any number of trailing h-steps). These might be called single-return M-Paths. The path of length n=2 is ud. The paths of length 3 are udh, uhd. The Paths of length 4 are uudd, udhh, uhdh and uhhd. A Motzkin Path can be chopped into subpaths of that type by splitting it at each u-step that starts from the horizontal line. [The exception is the path that consists entirely of h-steps.] The triangle of the Sequence Transform T(n,c) counts how many Motzkin Paths of length n which start with an u-step are concatenations of c of these single-return M-paths. T(n,1) are the single-return M-Paths. Row sums and column 1 are an INVERT transform pair.

			The triangle starts
      1
      2
      4     1
      8     4
     17    12     1
     38    32     6
     89    82    24     1
    216   208    80     8
    539   530   243    40    1
   1374  1364   702   160   10
   3562  3551  1975   564   60   1
   9360  9348  5484  1840  280  12
  24871 24858 15144  5716 1125  84  1
  66706 66692 41768 17208 4102 448 14
T(4,2)=1 counts udud.
T(5,1)=8 counts uuddh uudhd uuhdd udhhh uhudd uhdhh uhhdh uhhhd.
T(5,2)=4 counts ududh uduhd udhud uhdud.
T(2n,n) = 1 counts udududu... (ud repeated n times).

Cf. A086615 (column c=1), A002026 (row sums)

A348869 := proc(n,c)
    local g,x,y ;
    g := add( A086615(i)*x^(i+2),i=0..n) ;
    1/(1-y*g) ;
    coeftayl(%,x=0,n) ;
    coeftayl(%,y=0,c) ;
end proc:
seq(seq( A348869(n,c),c=1..n/2),n=2..10) ;

b[n_] := b[n] = If[n <= 3, 2^n, (3*(n+1)*b[n-1] + (n-4)*b[n-2] - 3*(n-1)*b[n-3])/(n+2)];
T[n_, c_] := Module[{g, x, y}, g = Sum[b[i]*x^(i+2), {i, 0, n}]; 1/(1-y*g) // SeriesCoefficient[#, {x, 0, n}]& // SeriesCoefficient[#, {y, 0, c}]&];
Table[T[n, c], {n, 2, 15}, {c, 1, n/2}] // Flatten (* Jean-François Alcover, Aug 12 2023, after Maple code *)

G.f.: 1/(1-y*g086615(x)) where g086615(x) = x^2 +2*x^3 +4*x^4 +8*x^5 +17*x^6 +....

A086614 Triangle read by rows, where T(n,k) is the coefficient of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x)^2 + xyf(x,y)^2.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 10, 12, 5, 5, 20, 42, 40, 14, 6, 35, 112, 180, 140, 42, 7, 56, 252, 600, 770, 504, 132, 8, 84, 504, 1650, 3080, 3276, 1848, 429, 9, 120, 924, 3960, 10010, 15288, 13860, 6864, 1430, 10, 165, 1584, 8580, 28028, 57330, 73920, 58344, 25740

Offset: 0

Paul D. Hanna, Jul 24 2003

			Rows:
{1},
{2, 1},
{3, 4,    2},
{4, 10,  12,    5},
{5, 20,  42,   40,   14},
{6, 35, 112,  180,  140,   42},
{7, 56, 252,  600,  770,  504,  132},
{8, 84, 504, 1650, 3080, 3276, 1848, 429}, ...

T(n,n) = A000108(n).

Cf. A086615 (antidiagonal sums), A086616 (row sums), A086617, A000292 (column 1), A277935 (column 2), A000580 (column 3 divided by 5), A000582 (column 4 divided by 14).

T := (n,k) -> `if`(k=0, n+1, binomial(2*k, k-1)*binomial(n+k+1, n-k)/k):
for n from 0 to 8 do seq(T(n,k), k=0..n) od; # Peter Luschny, Jan 26 2018

T(n,k) = binomial(2*k, k-1)*binomial(n+k+1, n-k) / k for k > 0. # Peter Luschny, Jan 26 2018

A136788 Triangle read by rows: A000012 * A107131.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 6, 7, 1, 1, 2, 6, 17, 11, 1, 1, 2, 6, 22, 41, 16, 1, 1, 2, 6, 22, 76, 86, 22, 1, 1, 2, 6, 22, 90, 226, 162, 29, 1, 1, 2, 6, 22, 90, 352, 582, 281, 37, 1

Offset: 0

Gary W. Adamson, Jan 21 2008

Row sums = A086615: (1, 2, 4, 8, 17, 38, ...).

A136787 = A107131 * A000012.

			First few rows of the triangle:
  1;
  1, 1;
  1, 2, 1;
  1, 2, 4,  1;
  1, 2, 6,  7,  1;
  1, 2, 6, 17, 11,  1;
  1, 2, 6, 22, 41, 16,  1;
  1, 2, 6, 22, 76, 86, 22, 1;
  ...

Cf. A086615, A136787.

A000012 * A107131 as infinite lower triangular matrices.

A364625 G.f. satisfies A(x) = 1/(1-x)^3 + x^2*A(x)^2.

Original entry on oeis.org

1, 3, 7, 16, 38, 95, 249, 678, 1901, 5451, 15906, 47066, 140868, 425657, 1296665, 3977684, 12276617, 38094013, 118768915, 371875752, 1168843808, 3686549845, 11664123048, 37011249678, 117750111763, 375529083267, 1200327617200, 3844662925222, 12338289374046

Offset: 0

Seiichi Manyama, Jul 30 2023

nonn

Cf. A000108, A086615, A162481, A360045.

Cf. A364626, A364627.

my(N=30, x='x+O('x^N)); Vec(2/((1-x)^3*(1+sqrt(1-4*x^2/(1-x)^3))))

G.f.: A(x) = 2 / ( (1-x)^3 * (1 + sqrt( 1 - 4*x^2/(1-x)^3 )) ).

a(n) = Sum_{k=0..floor(n/2)} binomial(n+k+2,3*k+2) * binomial(2*k,k) / (k+1).

A364622 G.f. satisfies A(x) = 1/(1-x)^2 + x^2*A(x)^4.

Original entry on oeis.org

1, 2, 4, 12, 45, 182, 779, 3480, 16005, 75234, 359893, 1746268, 8573477, 42511646, 212587561, 1070897000, 5429174465, 27679933778, 141829437174, 729972918876, 3772160853821, 19563615260102, 101797930474515, 531293155760840, 2780515192595481, 14588670579665882

Offset: 0

Seiichi Manyama, Jul 30 2023

nonn

Cf. A086615, A086631.

Table[Sum[Binomial[n + 4 k + 1, 6 k + 1]*Binomial[4 k, k]/(3 k + 1), {k, 0, Floor[n/2]}], {n, 0, 30}] (* Wesley Ivan Hurt, Jan 20 2024 *)

a(n) = sum(k=0, n\2, binomial(n+4*k+1, 6*k+1)*binomial(4*k, k)/(3*k+1));

a(n) = Sum_{k=0..floor(n/2)} binomial(n+4*k+1,6*k+1) * binomial(4*k,k) / (3*k+1).

A369691 G.f. satisfies A(x) = 1/(1-x)^3 + x^3*A(x)^3.

Original entry on oeis.org

1, 3, 6, 11, 24, 66, 196, 576, 1692, 5110, 15933, 50604, 161988, 521700, 1693362, 5541679, 18260055, 60487659, 201272437, 672550158, 2256204327, 7596059333, 25655943417, 86904524289, 295154911774, 1004906765178, 3429178160346, 11726499288028, 40178538608682

Offset: 0

Seiichi Manyama, Jan 29 2024

nonn

Cf. A364623, A364626.
Cf. A086615, A369693.
Cf. A364589.

a(n) = sum(k=0, n\3, binomial(n+3*k+2, n-3*k)*binomial(3*k, k)/(2*k+1));

a(n) = Sum_{k=0..floor(n/3)} binomial(n+3*k+2,n-3*k) * binomial(3*k,k) / (2*k+1).

A369693 G.f. satisfies A(x) = 1/(1-x)^4 + x^4*A(x)^4.

Original entry on oeis.org

1, 4, 10, 20, 36, 72, 220, 936, 4045, 15836, 56174, 187148, 616651, 2114448, 7717752, 29498000, 114243269, 437915876, 1650264874, 6149423732, 22909545269, 86129798600, 327872238092, 1260466647944, 4867739842821, 18801022899756, 72501445905366

Offset: 0

Seiichi Manyama, Jan 29 2024

nonn

Cf. A086615, A369691.

Cf. A364591.

a(n) = sum(k=0, n\4, binomial(n+8*k+3, n-4*k)*binomial(4*k, k)/(3*k+1));

a(n) = Sum_{k=0..floor(n/4)} binomial(n+8*k+3,n-4*k) * binomial(4*k,k) / (3*k+1).

cp's OEIS Frontend

A182013 Triangle of partial sums of Motzkin numbers.

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A348869 Triangle T(n,c) counting Motzkin Paths of length n with c sections starting with an up-step at level 0.

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A086614 Triangle read by rows, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x)^2 + xy*f(x,y)^2.

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A136788 Triangle read by rows: A000012 * A107131.

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A364625 G.f. satisfies A(x) = 1/(1-x)^3 + x^2*A(x)^2.

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PARI

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A364622 G.f. satisfies A(x) = 1/(1-x)^2 + x^2*A(x)^4.

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Mathematica

PARI

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A369691 G.f. satisfies A(x) = 1/(1-x)^3 + x^3*A(x)^3.

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PARI

Formula

A369693 G.f. satisfies A(x) = 1/(1-x)^4 + x^4*A(x)^4.

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PARI

Formula

A086614 Triangle read by rows, where T(n,k) is the coefficient of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x)^2 + xyf(x,y)^2.