A007477 -id:A007477 - cp's OEIS frontend

A143013 Number of Motzkin n-paths with two kinds of level steps one of which is a final step.

1, 2, 3, 7, 17, 43, 114, 310, 861, 2433, 6970, 20198, 59101, 174373, 518179, 1549545, 4659399, 14079553, 42732230, 130208246, 398174723, 1221573603, 3758835953, 11597578995, 35872937745, 111216324015, 345539568900, 1075693015920

Offset: 0

Michael Somos, Jul 15 2008

nonn

Hankel transform is the (4,-5) Somos-4 variant A171422. - Paul Barry, Dec 08 2009

			A = 1 + (L + F) + (LL + LF + UD) + (LLL + LLF + LUD + UDL + UDF + ULD + UFD) + ...
G.f. = 1 + 2*x + 3*x^2 + 7*x^3 + 17*x^4 + 43*x^5 + 114*x^6 + 310*x^7 + 861*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.

Cf. A000108, A007477, A171422.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-x -Sqrt(1-2*x-3*x^2-4*x^3))/(2*x^2) )); // G. C. Greubel, Feb 26 2019

CoefficientList[Series[(1-x -Sqrt[1-2*x-3*x^2-4*x^3])/(2*x^2), {x, 0, 30}], x] (* G. C. Greubel, Feb 26 2019 *)

a(n):=sum(sum((binomial(i-1,k-1)*binomial(k,n-k-i+2)*binomial(k+i-2,i-1))/k,k,1,n-i+2),i,0,n+2); /* Vladimir Kruchinin, May 06 2018 */

{a(n) = if(n<0, 0, polcoeff( (1 - x - sqrt(1 - 2*x - 3*x^2 - 4*x^3 + x^3*O(x^n))) / (2*x^2), n))}

x='x+O('x^30); Vec((1-x-(1-2*x-3*x^2-4*x^3)^(1/2))/(2*x^2)) \\ Altug Alkan, May 06 2018

((1-x -sqrt(1-2*x-3*x^2-4*x^3))/(2*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019

Words on alphabet {U,D,L,F} of length n where U is upstep, D is downstep, L and F are level steps and F can only be immediately followed by D or end of word with defining equation A = 1 + F + LA + UADA.
When convolved with itself yields first difference shifted left one place.
G.f. A(x) satisfies: A(x) = 1 + x + A(x)*x + (A(x)*x)^2.
G.f.: (1+x) / (1-x -(x^2 + x^3) / (1-x -(x^2 + x^3) / (1-x -...))).
G.f.: (1 - x - sqrt(1 - 2*x - 3*x^2 - 4*x^3)) / (2*x^2).
Given an integer t >= 1 and initial values u = [a_0, a_1, ..., a_{t-1}], we may define an infinite sequence Phi(u) by setting a_n = a_{n-1} + a_0*a_{n-1} + a_1*a_{n-2} + ... + a_{n-2}*a_1 for n >= t. For example Phi([1]) is the Catalan numbers A000108. The present sequence is Phi([0,1,2]). - Gary W. Adamson, Oct 27 2008
Conjecture: (n+2)*a(n) -(2*n+1)*a(n-1) +3*(1-n)*a(n-2) +2*(5-2*n)*a(n-3)=0. - R. J. Mathar, Oct 25 2012
a(n) = Sum_{i=0..n+2} Sum_{k=1..n-i+2} C(i-1,k-1)*C(k,n-k-i+2)*C(k+i-2,i-1)/k. - Vladimir Kruchinin, May 06 2018

A354734 a(0) = a(1) = 1; a(n) = 3 * Sum_{k=0..n-2} a(k) * a(n-k-2).

Original entry on oeis.org

1, 1, 3, 6, 21, 54, 189, 558, 1944, 6210, 21681, 72576, 254988, 878850, 3112101, 10935000, 39030660, 139001346, 499808232, 1797731496, 6506661798, 23583173328, 85847830965, 313063862436, 1145325387114, 4197826175634, 15424343762184, 56774049331356, 209400739623054

Offset: 0

Ilya Gutkovskiy, Jun 04 2022

nonn

Cf. A005159, A007477, A354733, A354735, A354736.

a[0] = a[1] = 1; a[n_] := a[n] = 3 Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 28}]
nmax = 28; CoefficientList[Series[(1 - Sqrt[1 - 12 x^2 (1 + x)])/(6 x^2), {x, 0, nmax}], x]

G.f. A(x) satisfies: A(x) = 1 + x + 3 * (x * A(x))^2.
G.f.: (1 - sqrt(1 - 12 * x^2 * (1 + x))) / (6 * x^2).
a(n) ~ sqrt((2+3*r)*(1+r)) / (sqrt(Pi) * n^(3/2) * r^n), where r = 2*cos(arccos(1/8)/3)/3 - 1/3. - Vaclav Kotesovec, Jun 04 2022

A354735 a(0) = a(1) = 1; a(n) = 4 * Sum_{k=0..n-2} a(k) * a(n-k-2).

Original entry on oeis.org

1, 1, 4, 8, 36, 96, 416, 1312, 5504, 19200, 79168, 293888, 1203712, 4648448, 19027968, 75411456, 309487616, 1248411648, 5144133632, 21011775488, 86971449344, 358540509184, 1490753372160, 6189315784704, 25843660619776, 107902536122368, 452308820819968, 1897178275512320

Offset: 0

Ilya Gutkovskiy, Jun 04 2022

nonn

Cf. A007477, A052704, A151403, A354733, A354734, A354736.

a[0] = a[1] = 1; a[n_] := a[n] = 4 Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 27}]
nmax = 27; CoefficientList[Series[(1 - Sqrt[1 - 16 x^2 (1 + x)])/(8 x^2), {x, 0, nmax}], x]

G.f. A(x) satisfies: A(x) = 1 + x + 4 * (x * A(x))^2.
G.f.: (1 - sqrt(1 - 16 * x^2 * (1 + x))) / (8 * x^2).
a(n) ~ sqrt((2+3*r)*(1+r)) / (sqrt(Pi) * n^(3/2) * r^n), where r = 2*cos(arccos(-5/32)/3)/3 - 1/3. - Vaclav Kotesovec, Jun 04 2022

A354736 a(0) = a(1) = 1; a(n) = 5 * Sum_{k=0..n-2} a(k) * a(n-k-2).

Original entry on oeis.org

1, 1, 5, 10, 55, 150, 775, 2550, 12500, 46250, 219375, 875000, 4075000, 17071250, 78796875, 341100000, 1569350000, 6947531250, 31966000000, 143761750000, 662668906250, 3014440000000, 13932834296875, 63921914062500, 296358191406250, 1368603488281250, 6365085546875000

Offset: 0

Ilya Gutkovskiy, Jun 04 2022

nonn

Cf. A007477, A156058, A354733, A354734, A354735.

a[0] = a[1] = 1; a[n_] := a[n] = 5 Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 26}]
nmax = 26; CoefficientList[Series[(1 - Sqrt[1 - 20 x^2 (1 + x)])/(10 x^2), {x, 0, nmax}], x]

G.f. A(x) satisfies: A(x) = 1 + x + 5 * (x * A(x))^2.
G.f.: (1 - sqrt(1 - 20 * x^2 * (1 + x))) / (10 * x^2).
a(n) ~ sqrt((2+3*r)*(1+r)) / (sqrt(Pi) * n^(3/2) * r^n), where r = 2*cos(arccos(-13/40)/3)/3 - 1/3. - Vaclav Kotesovec, Jun 04 2022

A366554 G.f. A(x) satisfies A(x) = 1 + x + x^4*A(x)^2.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 1, 0, 2, 6, 6, 2, 5, 20, 30, 20, 19, 70, 140, 140, 112, 266, 630, 840, 762, 1176, 2814, 4620, 5049, 6204, 12936, 24156, 31460, 36894, 63492, 123552, 185471, 228800, 338910, 634920, 1050686, 1411410, 1944800, 3354780, 5820256, 8513804, 11644490

Offset: 0

Seiichi Manyama, Oct 13 2023

nonn

Cf. A007477, A025227, A248100.
Cf. A366556, A366558.
Cf. A366589.

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

G.f.: A(x) = 2*(1+x) / (1+sqrt(1-4*x^4*(1+x))).
a(n) = Sum_{k=0..floor(n/4)} binomial(k+1,n-4*k) * binomial(2*k,k)/(k+1).
a(n) = A366589(n) + A366589(n-1).

A307374 G.f. A(x) satisfies: A(x) = 1 + x - x^2*A(x)^2.

Original entry on oeis.org

1, 1, -1, -2, 1, 6, 1, -18, -16, 50, 93, -112, -428, 98, 1713, 936, -6004, -8382, 17512, 47608, -33826, -221936, -36335, 892164, 862666, -3051022, -6076072, 8026380, 32247334, -8222288, -144487267, -81500652, 555489738, 801700858, -1751543424, -4898513044

Offset: 0

Ilya Gutkovskiy, Apr 06 2019

sign

			G.f.: A(x) = 1 + x - x^2 - 2*x^3 + x^4 + 6*x^5 + x^6 - 18*x^7 - 16*x^8 + 50*x^9 + 93*x^10 - 112*x^11 - 428*x^12 + ...

Cf. A007477, A104565.

terms = 35; A[] = 0; Do[A[x] = 1 + x - x^2 A[x]^2 + O[x]^(terms + 1) // Normal, {terms + 1}]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = -Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 35}]

a(0) = a(1) = 1; a(n+2) = -Sum_{k=0..n} a(k)*a(n-k).

A349014 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x) / (1 - x) + x^2 * A(x)^2.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 47, 111, 270, 663, 1656, 4174, 10636, 27308, 70651, 183902, 481436, 1266515, 3346793, 8879116, 23642034, 63156917, 169222939, 454660940, 1224650739, 3306338583, 8945780742, 24252558183, 65872671839, 179228552638, 488443704486

Offset: 0

Ilya Gutkovskiy, Nov 05 2021

nonn

Cf. A007477, A215973, A349015, A349016.

nmax = 30; A[] = 0; Do[A[x] = 1 + x + x^2 A[x]/(1 - x) + x^2 A[x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[a[k] (1 + a[n - k - 2]), {k, 0, n - 2}]; Table[a[n], {n, 0, 30}]

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} a(k) * (1 + a(n-k-2)).

a(n) ~ sqrt(1/r + (2-r)*s/(1-r)^2 + 2*s^2) / (2*sqrt(Pi)*n^(3/2)*r^n), where r = 0.3495518575342322867499973927570340375314361958565... and s = 3.323404276086477625771682790702806844309937221726... are real roots of the system of equations 1 + r + r^2*s*(1/(1-r) + s) = s, r^2*(1/(1-r) + 2*s) = 1. - Vaclav Kotesovec, Nov 06 2021

A349015 G.f. A(x) satisfies: A(x) = 1 + x * A(x) / (1 - x) - x * A(x)^2.

Original entry on oeis.org

1, 0, 1, 0, 2, -1, 5, -6, 16, -28, 62, -125, 267, -565, 1213, -2618, 5686, -12418, 27248, -60048, 132848, -294930, 656878, -1467257, 3286219, -7378239, 16603459, -37441989, 84599855, -191501531, 434224405, -986161958, 2243009870, -5108859820, 11651743902

Offset: 0

Ilya Gutkovskiy, Nov 05 2021

sign

Cf. A000108, A007477, A215973, A349014, A349016.

nmax = 34; A[] = 0; Do[A[x] = 1 + x A[x]/(1 - x) - x A[x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[a[k] (1 - a[n - k - 1]), {k, 0, n - 1}]; Table[a[n], {n, 0, 34}]

a(0) = 1; a(n) = Sum_{k=0..n-1} a(k) * (1 - a(n-k-1)).
a(n) = 1 - Sum_{k=0..n-1} (-1)^k * A007477(k).
a(n) ~ 3^(1 + n) * (1/((1 - 2/(19 - 3*sqrt(33))^(1/3) - (1/2)*(19 - 3*sqrt(33))^(1/3))^n * ((19 - 3*sqrt(33))^(1/6)*(2 + (19 - 3*sqrt(33))^(1/3))^2 * n^(3/2) * sqrt(((-1951699 + 339747*sqrt(33))*Pi) / (-70717234 + 12310290*sqrt(33) + (19 - 3*sqrt(33))^(2/3) * (-3903398 + 679494*sqrt(33)) + (19 - 3*sqrt(33))^(1/3) * (-35358617 + 6155145*sqrt(33))))))). - Vaclav Kotesovec, Nov 17 2021

A196182 Triangle T(n,k), read by rows, given by (0,1,1,0,1,1,0,1,1,0,1,1,0,...) DELTA (1,0,0,1,0,0,1,0,0,1,0,0,1,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 6, 3, 1, 0, 8, 17, 12, 4, 1, 0, 16, 46, 44, 20, 5, 1, 0, 32, 120, 150, 90, 30, 6, 1, 0, 64, 304, 482, 370, 160, 42, 7, 1, 0, 128, 752, 1476, 1412, 770, 259, 56, 8, 1, 0, 256, 1824, 4344, 5068, 3402, 1428, 392, 72, 9, 1

Offset: 0

Philippe Deléham, Oct 28 2011

Row sums are A000108 ; diagonal sums are A005043, antidiagonal sums are A007477.

			Triangle begins :
1
0, 1
0, 1, 1
0, 2, 2, 1
0, 4, 6, 3, 1
0, 8, 17, 12, 4, 1
0, 16, 46, 44, 20, 5, 1

Cf. A084938, A175136, A198379

A318122 a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} gcd(a(k), a(n-k-2)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 6, 8, 8, 13, 10, 16, 18, 21, 20, 28, 20, 30, 50, 36, 26, 67, 58, 48, 46, 72, 64, 76, 56, 93, 72, 96, 96, 138, 126, 112, 102, 160, 114, 160, 150, 144, 126, 128, 118, 273, 190, 252, 130, 230, 180, 260, 248, 312, 212, 208, 320, 422, 460, 296, 452, 493, 260, 436, 280

Offset: 0

Ilya Gutkovskiy, Aug 18 2018

nonn

N. J. A. Sloane, Transforms

Cf. A007464, A007477, A318123.

a[0] = a[1] = 1; a[n_] := a[n] = Sum[GCD[a[k], a[n - k - 2]], {k, 0, n - 2}]; Table[a[n], {n, 0, 65}]

cp's OEIS Frontend

A143013 Number of Motzkin n-paths with two kinds of level steps one of which is a final step.

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A354734 a(0) = a(1) = 1; a(n) = 3 * Sum_{k=0..n-2} a(k) * a(n-k-2).

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A354735 a(0) = a(1) = 1; a(n) = 4 * Sum_{k=0..n-2} a(k) * a(n-k-2).

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A354736 a(0) = a(1) = 1; a(n) = 5 * Sum_{k=0..n-2} a(k) * a(n-k-2).

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A366554 G.f. A(x) satisfies A(x) = 1 + x + x^4*A(x)^2.

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PARI

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A307374 G.f. A(x) satisfies: A(x) = 1 + x - x^2*A(x)^2.

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A349014 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x) / (1 - x) + x^2 * A(x)^2.

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A349015 G.f. A(x) satisfies: A(x) = 1 + x * A(x) / (1 - x) - x * A(x)^2.

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A196182 Triangle T(n,k), read by rows, given by (0,1,1,0,1,1,0,1,1,0,1,1,0,...) DELTA (1,0,0,1,0,0,1,0,0,1,0,0,1,...) where DELTA is the operator defined in A084938.

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