A143927 -id:A143927 - cp's OEIS frontend

A027908 a(n) = T(2*n, n), T given by A027907.

1, 2, 10, 50, 266, 1452, 8074, 45474, 258570, 1481108, 8533660, 49402850, 287134346, 1674425300, 9792273690, 57407789550, 337281021450, 1985342102964, 11706001102180, 69124774458092, 408737856117916, 2419833655003752, 14341910428953018, 85087759173024870

Offset: 0

Clark Kimberling

nonn

Central terms of the triangle in A111808. - Reinhard Zumkeller, Aug 17 2005

Number of paths of semilength n starting at (0,0) and ending on the X-axis using steps (1,1), (1,-1) and (1,3). - David Scambler, Jun 21 2013

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A027907, A027913, A143927, A344396, A370159, A370160.

ogf := series( RootOf( (144*x^2+140*x-27)*g^4+(18-12*x)*g^2+8*g+1, g), x=0, 20); # Mark van Hoeij, Nov 16 2011
a := n -> simplify(GegenbauerC(n, -2*n, -1/2)):
seq(a(n), n=0..23); # Peter Luschny, May 09 2016

Table[Binomial[4 n, n] Hypergeometric2F1[-3 n, -n, 1/2 - 2 n, 1/4], {n, 0, 20}] (* or *) Table[GegenbauerC[3 n, -2 n, -1/2] + KroneckerDelta[n], {n, 0, 20}] (* Vladimir Reshetnikov, May 07 2016 *)

makelist(ultraspherical(n,-2*n,-1/2),n,0,12); /* Emanuele Munarini, Oct 18 2016 */

G.f.: -(g^2+g+1)/(3*g^2+g-1) where g = x times the g.f. of A143927. - Mark van Hoeij, Nov 16 2011
a(n) = GegenbauerC(n, -2*n, -1/2). - Peter Luschny, May 09 2016
From Peter Bala, Jan 26 2020: (Start)
a(n) = [x^(2*n)](1 + x^2 + x^4)^(2*n).
a(n) = Sum_{k = 0..floor(n/2)} C(2*n, n-k)*C(n-k, k).
a(n) = C(2*n,n) * hypergeom([-n/2, (1 - n)/2], [n + 1], 4)
Conjectural: a(n*p^k) == a(n*p^(k-1)) ( mod p^(2*k) ) for all primes p >= 5 and positive integers n and k. (End)
From Peter Bala, Aug 03 2023: (Start)
P-recursive: 3*n*(13*n - 17)*(3*n - 1)*(3*n - 2)*a(n) = 2*(2*n - 1)*(455*n^3 - 1050*n^2 + 691*n - 120)*a(n-1) + 36*(n - 1)*(13*n - 4)*(2*n - 1)*(2*n - 3)*a(n-2) with a(0) = 1 and a(1) = 2.
exp(Sum_{n >= 0} a(n)*x^n/n) = 1 + 2*x + 7*x^2 + 28*x^3 + 123*x^4 + ... is the g.f. of A143927.
a(n) = 2*A344396(n-1) for n >= 1. (End)

A365128 G.f. satisfies A(x) = (1 + xA(x)(1 + x*A(x)))^3.

Original entry on oeis.org

1, 3, 15, 88, 567, 3876, 27607, 202653, 1522365, 11647038, 90435804, 710855544, 5645365576, 45228648078, 365109237801, 2966862631856, 24248879149005, 199213507774365, 1644138419038500, 13625326165675698, 113336685917785332, 945931091151789808

Offset: 0

Seiichi Manyama, Aug 23 2023

nonn

Cf. A001006, A143927.

Cf. A019497, A255673.

A365128 := proc(n)
    add(binomial(3*(n+1),k) * binomial(k,n-k),k=0..n) ;
    %/(n+1) ;
end proc:
seq(A365128(n),n=0..70) ; # R. J. Mathar, Dec 04 2023

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

If g.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x))^s)^t, then a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(t*(n+1),k) * binomial(s*k,n-k).
D-finite with recurrence 205*(5*n+6)*(5*n+2)*(5*n+3)*(5*n+4)*(n+1)*a(n) +9*(-5948592*n^5+11369145*n^4 -5182620*n^3 -351495*n^2+204302*n-6560) *a(n-1) +243*(-801282*n^5 +14391105*n^4 -55889790*n^3 +90254895*n^2 -66199848*n +18182560)*a(n-2) +6561*(3*n-5) *(3*n-4)*(93048*n^3 -579621*n^2 +1227037*n -878874)*a(n-3) +48715425*(n-3) *(3*n-4)*(3*n-7) *(3*n-5)*(3*n-8)*a(n-4)=0. - R. J. Mathar, Dec 04 2023
From Seiichi Manyama, Sep 20 2024: (Start)
G.f.: (1/x) * Series_Reversion( x / (1+x+x^2)^3 ).
G.f.: B(x)^3, where B(x) is the g.f. of A255673. (End)

A143926 G.f. satisfies: A(x) = 1 + xA(x)A(-x) + x^2A(x)^2A(-x)^2.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 7, 11, 28, 46, 123, 207, 572, 979, 2769, 4797, 13806, 24138, 70414, 123998, 365636, 647615, 1926505, 3428493, 10273870, 18356714, 55349155, 99229015, 300783420, 540807165, 1646828655, 2968468275, 9075674700

Offset: 0

Paul D. Hanna, Sep 06 2008

nonn

Bisections form A006605 and A143927;
A006605 is the number of modes of connections of 2n points and
A143927 is the self-convolution of A006605.

			G.f. A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 7*x^6 + 11*x^7 +...
A(x)*A(-x) = 1 + x^2 + 3*x^4 + 11*x^6 + 46*x^8 + 207*x^10 + 979*x^12 +...
A(x)^2*A(-x)^2 = 1 + 2*x^2 + 7*x^4 + 28*x^6 + 123*x^8 + 572*x^10 +...
A(x)^4*A(-x)^4 = 1 + 4*x^2 + 18*x^4 + 84*x^6 + 407*x^8 + 2028*x^10 +...
from this we see that if B(x^2) = A(x)*A(-x)
then B(x) = 1 + x*B(x)^2 + x^2*B(x)^4
and A(x) = 1 + x*B(x^2) + x^2*B(x^2)^2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Cf. A006605, A143927.

a[n_] := Module[{A = 1 + x, B}, For[i = 0, i <= n, i++, B = A*(A /. x -> -x); A = 1 + x*B + x^2*B^2 + O[x]^(n+1) // Normal]; SeriesCoefficient[A, {x, 0, n}]]; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Oct 22 2016, adapted from PARI *)

{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,B=A*subst(A,x,-x);A=1+x*B+x^2*B^2);polcoeff(A,n)}

Define B(x) by B(x^2) = A(x)*A(-x); then B(x) = 1 + x*B(x)^2 + x^2*B(x)^4 is the g.f. of A006605.
Recurrence: 3*n*(3*n+1)*(3*n+2)*(507*n^4 - 3575*n^3 + 8895*n^2 - 8953*n + 3054)*a(n) =  - 12*(4017*n^5 - 20319*n^4 + 31895*n^3 - 17595*n^2 + 2338*n + 384)*a(n-1) + 4*(n-2)*(17745*n^6 - 125125*n^5 + 331891*n^4 - 396335*n^3 + 173912*n^2 + 17532*n - 13140)*a(n-2) - 144*(n-3)*(n-2)*(312*n^3 - 988*n^2 + 407*n + 29)*a(n-3) + 144*(n-4)*(n-3)*(n-2)*(507*n^4 - 1547*n^3 + 1212*n^2 + 140*n - 72)*a(n-4). - Vaclav Kotesovec, Dec 21 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = sqrt(70/27+(26*sqrt(13))/27) = 2.4626418602647616787... is the root of the equation -144 - 140*d^2 + 27*d^4 = 0 and c = 2*sqrt((5+1/sqrt(13))/3)/3 = 0.88421131194123... if n is even, and c = sqrt(1+11/sqrt(13))/3 = 0.670890873659690... if n is odd. - Vaclav Kotesovec, Dec 21 2013

A371574 G.f. satisfies A(x) = ( 1 + xA(x)^(5/2) (1 + x*A(x)) )^2.

Original entry on oeis.org

1, 2, 13, 106, 986, 9902, 104641, 1146654, 12910674, 148462310, 1736178005, 20584835962, 246874102771, 2989580399330, 36504669373240, 448960388422126, 5556453433915920, 69150493021938224, 864833621158491876, 10863849369160145222, 137011477676531989664

Offset: 0

Seiichi Manyama, Mar 28 2024

nonn

Cf. A000108, A143927, A365153, A368961, A371575.

Cf. A365186.

a(n, r=2, s=1, t=5, u=2) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r).

G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A365186.

A027913 T(n,[ n/2 ]), T given by A027907.

Original entry on oeis.org

1, 1, 2, 3, 10, 15, 50, 77, 266, 414, 1452, 2277, 8074, 12727, 45474, 71955, 258570, 410346, 1481108, 2355962, 8533660, 13599915, 49402850, 78855339, 287134346, 458917850, 1674425300, 2679183405, 9792273690, 15683407785

Offset: 0

Clark Kimberling

nonn

The median coefficient in the expansion of (1 + x + x^2)^n. - Vladimir Reshetnikov, Nov 21 2020

Robert Israel, Table of n, a(n) for n = 0..2550

Cf. A027907, A027908.

seq(simplify(GegenbauerC(floor(n/2),-n,-1/2)), n=0..100); # Robert Israel, Oct 20 2016

Table[GegenbauerC[Floor[n/2], -n, -1/2] + KroneckerDelta[n, 0], {n, 0,
100}] (* Emanuele Munarini, Oct 20 2016 *)

makelist(ultraspherical(floor(n/2),-n,-1/2),n,0,12); /* Emanuele Munarini, Oct 18 2016 */

a(n) = GegenbauerC(floor(n/2), -n, -1/2). - Emanuele Munarini, Oct 18 2016

G.f.: g(t) = (1+(t+t^2)*A(t^2)+t^4*A(t^2)^2)/(1-t^2*A(t^2)-3*t^4*A(t^2)^2), where A(t) is the g.f. of A143927 and satisfies A(t) = [1 + x*A(t) + t^2*A(t)^2]^2. - Emanuele Munarini, Oct 20 2016

A369477 Expansion of (1/x) * Series_Reversion( x / ((1+x) * (1+x+x^2)^2) ).

Original entry on oeis.org

1, 3, 14, 77, 464, 2964, 19717, 135131, 947549, 6765642, 49022225, 359545750, 2664127354, 19913283809, 149968276974, 1136856855549, 8668000962927, 66428474900907, 511414514214628, 3953420853213504, 30674783555852576, 238808419235022293, 1864869207177530320

Offset: 0

Seiichi Manyama, Jan 23 2024

nonn

Index entries for reversions of series

Cf. A106228, A369479.
Cf. A143927, A369478.
Cf. A369440.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)*(1+x+x^2)^2))/x)

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

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

A378292 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(2n+k,r) binomial(r,n-r)/(2*n+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 11, 0, 1, 4, 12, 28, 46, 0, 1, 5, 18, 52, 123, 207, 0, 1, 6, 25, 84, 240, 572, 979, 0, 1, 7, 33, 125, 407, 1155, 2769, 4797, 0, 1, 8, 42, 176, 635, 2028, 5733, 13806, 24138, 0, 1, 9, 52, 238, 936, 3276, 10332, 29136, 70414, 123998, 0

Offset: 0

Seiichi Manyama, Nov 21 2024

			Square array begins:
  1,   1,    1,    1,     1,     1,     1, ...
  0,   1,    2,    3,     4,     5,     6, ...
  0,   3,    7,   12,    18,    25,    33, ...
  0,  11,   28,   52,    84,   125,   176, ...
  0,  46,  123,  240,   407,   635,   936, ...
  0, 207,  572, 1155,  2028,  3276,  4998, ...
  0, 979, 2769, 5733, 10332, 17140, 26860, ...

Columns k=0..2 give A000007, A006605, A143927.

Cf. A009766, A026300, A378289, A378290, A378291.

T(n, k, t=2, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(2/k) * (1 + x * A_k(x)^(2/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A006605.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+1) + x^2 * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+1) + T(n-2,k+3) for n > 1.

A369478 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x+x^2)^2) ).

Original entry on oeis.org

1, 4, 24, 170, 1320, 10868, 93197, 823484, 7445184, 68545882, 640446224, 6057249180, 57878746750, 557903174040, 5418441862824, 52971933934834, 520869559359424, 5147999004530720, 51113415228327827, 509583784051748692, 5099262428810825568

Offset: 0

Seiichi Manyama, Jan 23 2024

nonn

Index entries for reversions of series

Cf. A219537, A369480.
Cf. A143927, A369477.
Cf. A369441.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2*(1+x+x^2)^2))/x)

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

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

A371575 G.f. satisfies A(x) = ( 1 + xA(x)^3 (1 + x*A(x)) )^2.

Original entry on oeis.org

1, 2, 15, 144, 1587, 18942, 238301, 3111788, 41779164, 573127760, 7998164674, 113189243386, 1620583793262, 23431706243230, 341654376602948, 5017986762425680, 74170837061591036, 1102479579201183898, 16469074050937364044, 247115476148847822586

Offset: 0

Seiichi Manyama, Mar 28 2024

nonn

Cf. A000108, A143927, A365153, A368961, A371574.

a(n, r=2, s=1, t=6, u=2) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r).

A027910 T(2n,n-2), T given by A027907.

Original entry on oeis.org

1, 6, 36, 210, 1221, 7098, 41328, 241128, 1409895, 8260934, 48497064, 285219090, 1680166215, 9912297150, 58558256496, 346371955776, 2051126447742, 12158963346852, 72147074769640, 428476010502582, 2546776668682323, 15149061841758174, 90175327717962024

Offset: 2

Clark Kimberling

nonn

a(n) is also the number of lattice paths from (0,0) to (2n-1,n-2) taking north and east steps avoiding north^{>=3}. - Shanzhen Gao, Apr 20 2010

Alois P. Heinz, Table of n, a(n) for n = 2..500

a:= proc(n) option remember; `if`(n<3, n*(n-1)/2,
      (14*(2*n-1)*(65*n^3-120*n^2+37*n+6) *a(n-1)
      +36*(n-1)*(2*n-1)*(2*n-3)*(13*n+2) *a(n-2))/
      (3*(13*n-11)*(n-2)*(3*n+2)*(3*n+1)))
    end:
seq(a(n), n=2..25);  # Alois P. Heinz, Aug 07 2013

a(n) = Sum_{i=0..floor((2*n-3)/2)} C(2*n,n-2-i)*C(n-2-i,i). Shanzhen Gao, Apr 20 2010
G.f.: -g^2*(g^2+g+1)/(3*g^2+g-1) where g = x times the g.f. of A143927. - Mark van Hoeij, Nov 16 2011
a(n) ~ sqrt((221-29*sqrt(13))/78) * ((70+26*sqrt(13))/27)^n/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 25 2014

cp's OEIS Frontend

A027908 a(n) = T(2*n, n), T given by A027907.

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A365128 G.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x)))^3.

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A143926 G.f. satisfies: A(x) = 1 + x*A(x)*A(-x) + x^2*A(x)^2*A(-x)^2.

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A371574 G.f. satisfies A(x) = ( 1 + x*A(x)^(5/2) * (1 + x*A(x)) )^2.

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A027913 T(n,[ n/2 ]), T given by A027907.

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Maxima

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A369477 Expansion of (1/x) * Series_Reversion( x / ((1+x) * (1+x+x^2)^2) ).

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A378292 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(2*n+k,r) * binomial(r,n-r)/(2*n+k) for k > 0.

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A369478 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x+x^2)^2) ).

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A365128 G.f. satisfies A(x) = (1 + xA(x)(1 + x*A(x)))^3.

A143926 G.f. satisfies: A(x) = 1 + xA(x)A(-x) + x^2A(x)^2A(-x)^2.

A371574 G.f. satisfies A(x) = ( 1 + xA(x)^(5/2) (1 + x*A(x)) )^2.

A378292 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(2n+k,r) binomial(r,n-r)/(2*n+k) for k > 0.

A371575 G.f. satisfies A(x) = ( 1 + xA(x)^3 (1 + x*A(x)) )^2.