A091147 -id:A091147 - cp's OEIS frontend

A084605 G.f.: 1/(1-2x-15x^2)^(1/2); also, a(n) is the central coefficient of (1+x+4x^2)^n.

1, 1, 9, 25, 145, 561, 2841, 12489, 60705, 281185, 1353769, 6418809, 30917041, 148331665, 716698425, 3462260265, 16786700865, 81464917185, 396215601225, 1929237099225, 9408084660945, 45928695279345, 224476389327705

Offset: 0

Paul D. Hanna, Jun 01 2003

nonn

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U (or D) steps come in four colors. - N-E. Fahssi, Mar 30 2008
Ignoring initial term, equals the logarithmic derivative of A091147. - Paul D. Hanna, Dec 08 2018
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. - Peter Bala, Jan 10 2022

Seiichi Manyama, Table of n, a(n) for n = 0..1433 (terms 0..200 from Vincenzo Librandi)
Hacène Belbachir, Abdelghani Mehdaoui and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 96.
Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091.

Cf. A002426, A084600-A084604, A084606-A084615.

Cf. A322240 (a(n)^2), A091147.

a := n -> simplify(2^n*GegenbauerC(n,-n, -1/4)):
seq(a(n), n=0..22); # Peter Luschny, May 08 2016

Table[n!*SeriesCoefficient[E^x*BesselI[0,4*x],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)
a[n_] := Hypergeometric2F1[1/2 - n/2, -n/2, 1, 16];
Table[a[n], {n, 0, 22}] (* Peter Luschny, Mar 18 2018 *)

for(n=0,30,t=polcoeff((1+x+4*x^2)^n,n,x); print1(t","))
for(n=0,20,print1(a(n),", "))

{a(n) = sum(k=0,n, (-3)^(n-k)*2^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Dec 09 2018

E.g.f.: exp(x)*BesselI(0, 4*x). - Vladeta Jovovic, Aug 20 2003
a(n) is also the central coefficient of (4+x+x^2)^n; a(n) = Sum_{k=0..n} 3^(n-k) C(n,k) T(k,n), where T(k,n) is the triangle of trinomial coefficients = Coefficient of x^n of (1+x+x^2)^k : A027907. - N-E. Fahssi, Mar 30 2008
a(n) = (1/Pi)*integral(x=-2..2, (2*x+1)^n/sqrt((2-x)*(2+x))). - Peter Luschny, Sep 12 2011
D-finite with recurrence a(n+2) = ((2*n+3)*a(n+1) + 15*(n+1)*a(n))/(n+2); a(0)=a(1)=1 - Sergei N. Gladkovskii, Aug 01 2012
a(n) ~ 5^(n+1/2)/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Oct 14 2012
a(n) = 2^n*GegenbauerC(n, -n, -1/4). - Peter Luschny, May 08 2016
a(n) = hypergeom([1/2 - n/2, -n/2], [1], 16). - Peter Luschny, Mar 18 2018
a(n) = Sum_{k=0..n} (-3)^(n-k) * 2^k * binomial(n,k)*binomial(2*k,k). - Paul D. Hanna, Dec 09 2018
a(n) = Sum_{k=0..n} 5^(n-k) * (-2)^k * binomial(n,k)*binomial(2*k,k). - Seiichi Manyama, May 01 2019
a(n) = (1/4)^n * Sum_{k=0..n} (-3)^k * 5^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

A217275 Expansion of 2/(1-x+sqrt(1-2x-27x^2)).

Original entry on oeis.org

1, 1, 8, 22, 141, 561, 3291, 15583, 88691, 459187, 2599570, 14136200, 80391235, 450046143, 2579291352, 14710321998, 85002979083, 491050703739, 2859262171872, 16674374605722, 97747766045679, 574231140306699, 3385974360904227, 20009363692187115, 118582649963026677

Offset: 0

Vaclav Kotesovec, Sep 29 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001006 (z=1), A025235 (z=2), A025237 (z=3), A091147 (z=4), A091148 (z=5), A091149 (z=6).

Table[SeriesCoefficient[2/(1-x+Sqrt[1-2*x-27*x^2]),{x,0,n}],{n,0,25}]
Table[Sum[Binomial[n,2k]*Binomial[2k,k]*7^k/(k+1),{k,0,n}],{n,0,25}]

Generally for G.f. = 2/(1-x+sqrt(1-2x-(4*z-1)*x^2)) is asymptotic
a(n) ~ (1+2*sqrt(z))^(n+3/2)/(2*sqrt(Pi)*z^(3/4)*n^(3/2)); here we have the case z=7.
D-finite with recurrence: (n+2)*a(n)=(2*n+1)*a(n-1)+(4*z-1)*(n-1)*a(n-2);; here with z=7.
G.f.: 1/(1 - x - 7*x^2/(1 - x - 7*x^2/(1 - x - 7*x^2/(1 - x - 7*x^2/(1 - ....))))), a continued fraction. - Ilya Gutkovskiy, May 26 2017

A306684 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2x + (1-4k)*x^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 7, 9, 1, 1, 1, 5, 10, 21, 21, 1, 1, 1, 6, 13, 37, 61, 51, 1, 1, 1, 7, 16, 57, 121, 191, 127, 1, 1, 1, 8, 19, 81, 201, 451, 603, 323, 1, 1, 1, 9, 22, 109, 301, 861, 1639, 1961, 835, 1

Offset: 0

Seiichi Manyama, May 06 2019

			Square array begins:
   1,   1,   1,    1,    1,    1,     1,     1, ...
   1,   1,   1,    1,    1,    1,     1,     1, ...
   1,   2,   3,    4,    5,    6,     7,     8, ...
   1,   4,   7,   10,   13,   16,    19,    22, ...
   1,   9,  21,   37,   57,   81,   109,   141, ...
   1,  21,  61,  121,  201,  301,   421,   561, ...
   1,  51, 191,  451,  861, 1451,  2251,  3291, ...
   1, 127, 603, 1639, 3445, 6231, 10207, 15583, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..7 give A000012, A001006, A025235, A025237, A091147, A091148, A091149, A217275.
Main diagonal gives A307906.
Cf. A000108, A107267, A247495, A307855.

T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[n, 2*j] * CatalanNumber[j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 12 2021 *)

A(n,k) is the coefficient of x^n in the expansion of 1/(n+1) * (1 + x + k*x^2)^(n+1).
A(n,k) = Sum_{j=0..floor(n/2)} k^j * binomial(n,j) * binomial(n-j,j)/(j+1) = Sum_{j=0..floor(n/2)} k^j * binomial(n,2*j) * A000108(j).
(n+2) * A(n,k) = (2*n+1) * A(n-1,k) - (1-4*k) * (n-1) * A(n-2,k).

A359364 Triangle read by rows. The Motzkin triangle, the coefficients of the Motzkin polynomials. M(n, k) = binomial(n, k) * CatalanNumber(k/2) if k is even, otherwise 0.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 0, 2, 1, 0, 10, 0, 10, 0, 1, 0, 15, 0, 30, 0, 5, 1, 0, 21, 0, 70, 0, 35, 0, 1, 0, 28, 0, 140, 0, 140, 0, 14, 1, 0, 36, 0, 252, 0, 420, 0, 126, 0, 1, 0, 45, 0, 420, 0, 1050, 0, 630, 0, 42, 1, 0, 55, 0, 660, 0, 2310, 0, 2310, 0, 462, 0

Offset: 0

Peter Luschny, Jan 09 2023

The generalized Motzkin numbers M(n, k) are a refinement of the Motzkin numbers M(n) (A001006) in the sense that they are coefficients of polynomials M(n, x) = Sum_{n..k} M(n, k) * x^k that take the value M(n) at x = 1. The coefficients of x^n are the aerated Catalan numbers A126120.
Variants are the irregular triangle A055151 with zeros deleted, A097610 with reversed rows, A107131 and A080159.
In the literature the name 'Motzkin triangle' is also used for the triangle A026300, which is generated from the powers of the generating function of the Motzkin numbers.

			Triangle M(n, k) starts:
[0] 1;
[1] 1, 0;
[2] 1, 0,  1;
[3] 1, 0,  3, 0;
[4] 1, 0,  6, 0,   2;
[5] 1, 0, 10, 0,  10, 0;
[6] 1, 0, 15, 0,  30, 0,   5;
[7] 1, 0, 21, 0,  70, 0,  35, 0;
[8] 1, 0, 28, 0, 140, 0, 140, 0,  14;
[9] 1, 0, 36, 0, 252, 0, 420, 0, 126, 0;

M. Artioli, G. Dattoli, S. Licciardi, and S. Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017, (Table 1 on p. 3).

Variants: A055151, A080159, A097610, A107131.
Columns M(n, 2*k): A000012, A000217, A034827, A000910, A088625, A088626, A213380.
Cf. A001006 (Motzkin numbers), A026300 (Motzkin gf. triangle), A126120 (aerated Catalan), A000108 (Catalan).
Cf. A138364, A107587, A002457, A002522, A025179, A025235, A056107, A014531, A023426, A091147, A189912, A343386, A343773, A359647, A359649.

CatalanNumber := n -> binomial(2*n, n)/(n + 1):
M := (n, k) -> ifelse(irem(k, 2) = 1, 0, CatalanNumber(k/2)*binomial(n, k)):
for n from 0 to 9 do seq(M(n, k), k = 0..n) od;
# Alternative, as coefficients of polynomials:
p := n -> hypergeom([(1 - n)/2, -n/2], [2], (2*x)^2):
seq(print(seq(coeff(simplify(p(n)), x, k), k = 0..n)), n = 0..9);
# Using the exponential generating function:
egf := exp(x)*BesselI(1, 2*x*t)/(x*t): ser:= series(egf, x, 11):
seq(print(seq(coeff(simplify(n!*coeff(ser, x, n)), t, k), k = 0..n)), n = 0..9);

from functools import cache
@cache
def M(n: int, k: int) -> int:
    if k %  2: return 0
    if n <  3: return 1
    if n == k: return (2 * (n - 1) * M(n - 2, n - 2)) // (n // 2 + 1)
    return (M(n - 1, k) * n) // (n - k)
for n in range(10): print([M(n, k) for k in range(n + 1)])

Let p(n, x) = hypergeom([(1 - n)/2, -n/2], [2], (2*x)^2).
p(n, 1) = A001006(n); p(n, sqrt(2)) = A025235(n); p(n, 2) = A091147(n).
p(2, n) = A002522(n); p(3, n) = A056107(n).
p(n, n) = A359649(n); 2^n*p(n, 1/2) = A000108(n+1).
M(n, k) = [x^k] p(n, x).
M(n, k) = [t^k] (n! * [x^n] exp(x) * BesselI(1, 2*t*x) / (t*x)).
M(n, k) = [t^k][x^n] ((1 - x - sqrt((x-1)^2 - (2*t*x)^2)) / (2*(t*x)^2)).
M(n, n) = A126120(n).
M(n, n-1) = A138364(n), the number of Motzkin n-paths with exactly one flat step.
M(2*n, 2*n) = A000108(n), the number of peakless Motzkin paths having a total of n up and level steps.
M(4*n, 2*n) = A359647(n), the central terms without zeros.
M(2*n+2, 2*n) = A002457(n) = (-4)^n * binomial(-3/2, n).
Sum_{k=0..n} M(n - k, k) = A023426(n).
Sum_{k=0..n} k * M(n, k) = 2*A014531(n-1) = 2*GegenbauerC(n - 2, -n, -1/2).
Sum_{k=0..n} i^k*M(n, k) = A343773(n), (i the imaginary unit), is the excess of the number of even Motzkin n-paths (A107587) over the odd ones (A343386).
Sum_{k=0..n} Sum_{j=0..k} M(n, j) = A189912(n).
Sum_{k=0..n} Sum_{j=0..k} M(n, n-j) = modified A025179(n).
For a recursion see the Python program.

A322241 G.f.: exp( Sum_{n>=1} A084605(n)^2 * x^n/n ), where A084605(n) is the central coefficient in (1 + x + 4*x^2)^n.

Original entry on oeis.org

1, 1, 41, 249, 6305, 77569, 1665321, 27724889, 574252417, 10958980929, 228679916905, 4671350051321, 99292476904609, 2107949882690241, 45658568907254505, 993562984208479193, 21876513296218002433, 484448162130512673665, 10812975015547281792937, 242647271141110287979513, 5477046865641884201456033

Offset: 0

Paul D. Hanna, Dec 08 2018

nonn

Compare to: exp( Sum_{n>=1} A084605(n) * x^n/n ) = (1-x - sqrt(1 - 2*x - 15*x^2))/(8*x^2), the g.f. of A091147.

Sequence A322240(n) = A084605(n)^2 has generating function 1 / AGM(1 + 15*x, sqrt((1 - 9*x)*(1 - 25*x)) ).

			G.f.: A(x) = 1 + x + 41*x^2 + 249*x^3 + 6305*x^4 + 77569*x^5 + 1665321*x^6 + 27724889*x^7 + 574252417*x^8 + 10958980929*x^9 + 228679916905*x^10 + ...
such that
log(A(x)) = x + 81*x^2/2 + 625*x^3/3 + 21025*x^4/4 + 314721*x^5/5 + 8071281*x^6/6 + 155975121*x^7/7 + 3685097025*x^8/8 + ... + A084605(n)^2 * x^n/n + ...
RELATED SERIES.
The g.f. of A084605 equals the series
1/sqrt(1 - 2*x - 15*x^2) = 1 + x + 9*x^2 + 25*x^3 + 145*x^4 + 561*x^5 + 2841*x^6 + 12489*x^7 + 60705*x^8 + 281185*x^9 + ... + A084605(n) * x^n/n + ...

Cf. A322240, A084605.

{a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, polcoeff(1/sqrt(1 - 2*x - 15*x^2 +x*O(x^m)), m)^2 *x^m/m)+x*O(x^n)), n))}
for(n=0,30,print1(a(n),", "))

A344557 Triangle read by rows, T(n, k) = 2^(n - k)*M(n, k, 1/2, 1/2), where M(n, k, x, y) is a generalized Motzkin recurrence. T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 5, 2, 1, 13, 11, 3, 1, 57, 36, 18, 4, 1, 201, 165, 70, 26, 5, 1, 861, 646, 339, 116, 35, 6, 1, 3445, 2863, 1449, 595, 175, 45, 7, 1, 14897, 12104, 6692, 2744, 950, 248, 56, 8, 1, 63313, 53769, 29772, 13236, 4686, 1422, 336, 68, 9, 1

Offset: 0

Peter Luschny, May 25 2021

The convolution triangle of A091147. - Peter Luschny, Oct 07 2022

			[0]     1;
[1]     1,     1;
[2]     5,     2,     1;
[3]    13,    11,     3,     1;
[4]    57,    36,    18,     4,    1;
[5]   201,   165,    70,    26,    5,    1;
[6]   861,   646,   339,   116,   35,    6,   1;
[7]  3445,  2863,  1449,   595,  175,   45,   7,  1;
[8] 14897, 12104,  6692,  2744,  950,  248,  56,  8, 1;
[9] 63313, 53769, 29772, 13236, 4686, 1422, 336, 68, 9, 1.

A091147 (first column), A344558 (row sums).

t := proc(n, k) option remember; if n = k then 1 elif k < 0 or n < 0 or k > n then 0 elif k = 0 then t(n-1, 0)/2 + t(n-1, 1) else t(n-1, k-1) + (1/2)*t(n-1, k) + t(n-1, k+1) fi end: T := (n, k) -> 2^(n-k) * t(n, k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
# Uses function PMatrix from A357368. Adds a row and column above and to the right.
PMatrix(10, n -> simplify(hypergeom([1/2-n/2, 1-n/2], [2], 16))); # Peter Luschny, Oct 07 2022

(* Uses function PMatrix from A357368 *)
nmax = 9;
M = PMatrix[nmax+2, HypergeometricPFQ[{1/2 - #/2, 1 - #/2}, {2}, 16]&];
T[n_, k_] := M[[n+2, k+2]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 29 2023, after Peter Luschny *)

The generalized Motzkin recurrence M(n, k, x, y) is defined as follows:

If k < 0 or n < 0 or k > n then 0 else if n = 0 then 1 else if k = 0 then x*M(n-1, 0, x, y) + M(n-1, 1, x, y). In all other cases M(n, k, x, y) = M(n-1, k-1, x, y) + y*M(n-1, k, x, y) + M(n-1, k+1, x, y).

A272868 Triangle read by rows, T(n,k) = 2^k*GegenbauerC(k,-n,-1/4), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 2, 9, 1, 3, 15, 25, 1, 4, 22, 52, 145, 1, 5, 30, 90, 285, 561, 1, 6, 39, 140, 495, 1206, 2841, 1, 7, 49, 203, 791, 2261, 6027, 12489, 1, 8, 60, 280, 1190, 3864, 11452, 27560, 60705, 1, 9, 72, 372, 1710, 6174, 20076, 54468, 134073, 281185

Offset: 0

Peter Luschny, May 08 2016

			Triangle starts:
                                1;
                              1, 1;
                            1, 2, 9;
                          1, 3, 15, 25;
                       1, 4, 22, 52, 145;
                     1, 5, 30, 90, 285, 561;
                 1, 6, 39, 140, 495, 1206, 2841;
             1, 7, 49, 203, 791, 2261, 6027, 12489;

Cf. A084605, A091147, A098520.

T := (n,k) -> simplify(2^k*GegenbauerC(k, -n, -1/4)):
for n from 0 to 9 do seq(T(n,k), k=0..n) od;

Table[If[n == 0, 1, 2^k GegenbauerC[k, -n, -1/4]], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 08 2016 *)

T(n,n) = A084605(n).
T(n,n-1) = A098520(n).
T(n+1,n)/(n+1) = A091147(n).

cp's OEIS Frontend

A084605 G.f.: 1/(1-2x-15x^2)^(1/2); also, a(n) is the central coefficient of (1+x+4x^2)^n.

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A217275 Expansion of 2/(1-x+sqrt(1-2*x-27*x^2)).

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A306684 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2*x + (1-4*k)*x^2)).

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A359364 Triangle read by rows. The Motzkin triangle, the coefficients of the Motzkin polynomials. M(n, k) = binomial(n, k) * CatalanNumber(k/2) if k is even, otherwise 0.

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A322241 G.f.: exp( Sum_{n>=1} A084605(n)^2 * x^n/n ), where A084605(n) is the central coefficient in (1 + x + 4*x^2)^n.

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A344557 Triangle read by rows, T(n, k) = 2^(n - k)*M(n, k, 1/2, 1/2), where M(n, k, x, y) is a generalized Motzkin recurrence. T(n, k) for 0 <= k <= n.

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A272868 Triangle read by rows, T(n,k) = 2^k*GegenbauerC(k,-n,-1/4), for n>=0 and 0<=k<=n.

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A217275 Expansion of 2/(1-x+sqrt(1-2x-27x^2)).

A306684 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2x + (1-4k)*x^2)).