A025227 -id:A025227 - cp's OEIS frontend

A379173 G.f. A(x) satisfies A(x) = (1 + x)/(1 - x*A(x))^2.

1, 3, 11, 53, 284, 1630, 9794, 60830, 387390, 2515892, 16599051, 110943779, 749603067, 5111606801, 35133394554, 243146923574, 1692918638012, 11850006727400, 83341778073920, 588646472669454, 4173607638548291, 29694593381322531, 211941668053441490, 1517087043428034420

Offset: 0

Seiichi Manyama, Dec 17 2024

nonn

Cf. A003169, A249924, A379174.

Cf. A025227, A379171.

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

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

A052727 Expansion of e.g.f. 1/2-1/2(1-4x-4*x^2)^(1/2).

Original entry on oeis.org

0, 1, 4, 24, 288, 4800, 103680, 2741760, 85800960, 3100446720, 127037030400, 5819550105600, 294727768473600, 16350861400473600, 986127353590579200, 64238655955009536000, 4495021381191204864000, 336249161369543245824000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Old name was: A simple context-free grammar in a labeled universe.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 683

spec := [S,{B=Prod(S,S),S=Union(B,Z,C),C=Prod(Z,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[1/2-1/2*(1-4*x-4*x^2)^(1/2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

Recurrence: {a(1)=1, a(2)=4, (-4*n^2+4)*a(n) +(-4*n-2)*a(n+1) +a(n+2) =0.
a(n) ~ sqrt(2-sqrt(2))* ((1+sqrt(2))/exp(1))^n * (2*n)^(n-1). - Vaclav Kotesovec, Sep 30 2013
a(n) = n!*A025227(n). - R. J. Mathar, Oct 18 2013

A068765 Generalized Catalan numbers 3xA(x)^2 -A(x) +1 -2*x = 0.

Original entry on oeis.org

1, 1, 6, 39, 270, 1962, 14796, 114831, 911574, 7368894, 60457428, 502162902, 4214515212, 35686162548, 304491863448, 2615468845311, 22598114065254, 196269877811574, 1712578870493316, 15005719955119698

Offset: 0

Wolfdieter Lang, Mar 04 2002

a(n)=K(3,3; n)/3 with K(a,b; n) defined in a comment to A068763.

Fung Lam, Table of n, a(n) for n = 0..1000

Cf. A000108, A068764, A068766-72, A025227-30.

CoefficientList[Series[(1-Sqrt[1-12*x*(1-2*x)])/(6*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

a(n)=(3^n)*p(n, -2/3) with the row polynomials p(n, x) defined from array A068763.
a(n+1)= 3*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-12*x*(1-2*x)))/(6*x).
Recurrence: (n+1)*a(n) = 24*(2-n)*a(n-2) + 6*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014
a(n) ~ sqrt(6+6*sqrt(3)) * (6+2*sqrt(3))^n / (6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014

A176703 Coefficients of a recursive polynomial based on Chaitin's S expressions: a(0)=1; a(1)=x; a(2)=1; a(n)=vector(a(n-1)).reverse(a(n-1)).

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 1, 4, 2, 2, 9, 8, 4, 2, 22, 24, 14, 8, 56, 70, 52, 24, 5, 146, 208, 176, 84, 30, 388, 624, 574, 320, 120, 14, 1048, 1876, 1868, 1184, 470, 112, 2869, 5648, 6088, 4236, 1900, 560, 42, 7942, 17040, 19804, 14928, 7560, 2492, 420, 22192, 51526, 64232

Offset: 0

Roger L. Bagula, Apr 24 2010

The result is an alternative way to expand s expressions as a binary rooted tree recursion.

			{1},
{0, 1},
{1, 0},
{2, 0, 1},
{4, 2, 2},
{9, 8, 4, 2},
{22, 24, 14, 8},
{56, 70, 52, 24, 5},
{146, 208, 176, 84, 30},
{388, 624, 574, 320, 120, 14},
{1048, 1876, 1868, 1184, 470, 112},
{2869, 5648, 6088, 4236, 1900, 560, 42},
{7942, 17040, 19804, 14928, 7560, 2492, 420},
{22192, 51526, 64232, 52208, 29190, 10864, 2520, 132},
{62510, 156128, 207808, 181320, 110260, 46256, 12684, 1584},
{177308, 473952, 670966, 625408, 410400, 190932, 59976, 11088, 429}

G. J. Chaitin, Algorithmic Information Theory, Cambridge Press, 1987, page 169

Cf. A025227, A025262, A072851, A124027

a[0] := 1; a[1] := x; a[2] = 1;
a[n_] := a[n] = Table[a[i], {i, 0, n - 1}].Table[a[n - 1 - i], {i, 0, n - 1}];
Table[ CoefficientList[a[n], x], {n, 0, 15}];
Flatten[%]

{T(n, k) = if( 2*k-1  > n, 0, polcoeff( polcoeff( ( 1 - sqrt( (1 - 2*x)^2 - 4*x^2 * (x + y - 2*x*y) + x^2*O(x^n))) / (2*x), n), k))} /* Michael Somos, Jan 09 2012 */

a(0)=1;a(1)=x;a(2)=1;
a(n)=vector(a(n-1)).reverse(a(n-1));
t(n,m)=coefficients(a(n) in x)
Let b(0) = b(2) = 1, b(1) = x, and b(n) = Sum_{i=1..n} b(i-1) * b(n-i) if n>2. Then T(n, k) = coefficient of x^k in b(n) where 0 <= k <= (n+1)/2.
G.f. A(x,y) satisfies A(x,y) = 1 - x * (1 - x - y + 2*x*y) + x * A(x,y)^2. - Michael Somos, Jan 09 2012
G.f.: ( 1 - sqrt( (1 - 2*x)^2 - 4*x^2 * (x + y - 2*x*y) )) / (2*x). - Michael Somos, Jan 09 2012
Row sums are A025262 if offset 0.

A290147 Expansion of (1-sqrt(1-8x-8x^2))/(4*x).

Original entry on oeis.org

1, 3, 12, 66, 408, 2712, 18912, 136488, 1010784, 7637664, 58650240, 456377664, 3590674176, 28516332288, 228297907200, 1840515987072, 14929020470784, 121749590032896, 997676696045568, 8210704762960896, 67835018440593408, 562407734010335232, 4677727530446635008

Offset: 0

R. J. Mathar, Jul 21 2017

nonn

By the application of enumerating Rota-Baxter word (not following the g.f.) the value at index 0 is set to a(0)=1.

Given y-2*y^2=x+x^2, expand y as a series in x, and then this sequence gives the coefficients: y=x+3*x^2+12*x^3+66*x^4+... (see PariGP code). - Robert Munafo, Oct 17 2024

Robert Israel, Table of n, a(n) for n = 0..1057
L. Guo and W. Y. Sit, Enumeration and generating functions of Rota-Baxter Words, Math. Comput. Sci. 4 (2010) 313-337, theorem 3.6 at z=2.

Cf. A025227.

f:= gfun:-rectoproc({8*n*a(n)+(12+8*n)*a(1+n)+(-3-n)*a(n+2), a(0) = 1, a(1) = 3},a(n),remember):
map(f, [$0..50]); # Robert Israel, Jul 21 2017

CoefficientList[Series[(1-Sqrt[1-8x-8x^2])/(4x),{x,0,30}],x] (* Harvey P. Dale, Feb 10 2018 *)

my(x='x+O('x^99)); Vec((1-sqrt(1-8*x-8*x^2))/(4*x)) \\ Altug Alkan, Jul 22 2017

my(y=x+O(x)); for(n=1,23,y=x+x^2+2*y^2); Vec(y) \\ Robert Munafo, Oct 17 2024

D-finite with recurrence (n+1)*a(n) +4*(-2*n+1)*a(n-1) +8*(-n+2)*a(n-2)=0. - R. J. Mathar, Jul 21 2017

a(n) ~ sqrt(3 - sqrt(6)) * 2^(n - 3/2) * (2 + sqrt(6))^(n+1) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2024

A337991 Triangle read by rows: T(n,m) = Sum_{i=1..n} C(n,i-m)C(n+m-i,i-1)C(n+m-i,m)/n, with T(0,0)=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 5, 4, 1, 4, 13, 15, 7, 1, 9, 35, 52, 36, 11, 1, 21, 96, 175, 160, 75, 16, 1, 51, 267, 576, 655, 415, 141, 22, 1, 127, 750, 1869, 2541, 2030, 952, 245, 29, 1, 323, 2123, 6000, 9492, 9156, 5488, 1988, 400, 37, 1, 835, 6046, 19107, 34476, 38976, 28476, 13356, 3852, 621, 46, 1

Offset: 0

Vladimir Kruchinin, Oct 06 2020

			Triangle begins as:
   1;
   1,   1;
   1,   2,   1;
   2,   5,   4,   1;
   4,  13,  15,   7,   1;
   9,  35,  52,  36,  11,   1;
  21,  96, 175, 160,  75,  16,  1;
  51, 267, 576, 655, 415, 141, 22,  1;
  ...

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A001006, A005773, A086246, A132894.
Diagonals include: A000124, A006008.
Sums include: A000007 (signed row), A019590 (signed diagonal), A025227 (row), A102407 (diagonal).

B:=Binomial;
A337991:= func< n,k | n eq 0 select 1 else (1/n)*(&+[B(n, j-k)*B(n+k-j, j-1)*B(n+k-j, k): j in [1..n]]) >;
[A337991(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 31 2024

T[0, 0] = 1; T[n_, m_] := Sum[Binomial[n, i - m] * Binomial[n + m - i, i - 1] * Binomial[n + m - i, m]/n, {i, 1, n}]; Table[T[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Amiram Eldar, Oct 06 2020 *)

T(n,m):=if m=n then 1 else if n=0 then 0 else sum(binomial(n,i-m)*binomial(n+m-i,i-1)*binomial(n+m-i,m),i,1,n)/n;

def A337991(n,k):
    b=binomial
    if n==0: return 1
    else: return (1/n)*sum(b(n, j-k)*b(n+k-j, j-1)*b(n+k-j, k) for j in range(1,n+1))
# SageMath
flatten([[A337991(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 31 2024

G.f.: ( 1 - x*(y-1)- sqrt(x^2*(y^2-2*y-3) - 2*x*(y+1) + 1) )/(2*x).
From G. C. Greubel, Oct 31 2024: (Start)
T(n, k) = binomial(n, 1-k)*binomial(n+k-1, k)*Hypergeometric3F2([1-n, (1 -n -k)/2, (2-n-k)/2], [2-k, 1-n-k], 4), with T(0, 0) = 1.
T(n, 0) = A086246(n+1).
T(n, n-1) = A000124(n-1), n >= 1.
T(n, n-2) = A006008(n-1), n >= 2.
T(n, n-3) = (1/72)*(n^4 -6*n^3 +47*n^2 -114*n +144)*binomial(n-1,2), n >= 3.
T(n, n-4) = (1/480)*(n-2)*(n^4 -8*n^3 +99*n^2 -332*n +960)*binomial(n-1,3), n >= 4.
Sum_{k=0..n} T(n, k) = A025227(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A102407(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A019590(n+1). (End)

A381937 G.f. A(x) satisfies A(x) = (1 + x) * B(x*A(x)), where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 2, 6, 35, 240, 1805, 14386, 119365, 1020136, 8918423, 79380514, 716911887, 6553219720, 60513355786, 563648995020, 5289485238552, 49963186247220, 474655663418546, 4532279676629700, 43473774550929628, 418706702628897708, 4047555977981218963

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A025227, A381787, A381940.

Cf. A001764, A346646, A365178.

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

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

a(n) = A365178(n) + A365178(n-1).

A381940 G.f. A(x) satisfies A(x) = (1 + x) * B(x*A(x)), where B(x) is the g.f. of A002293.

Original entry on oeis.org

1, 2, 7, 51, 440, 4170, 41921, 438972, 4736281, 52286520, 587774685, 6705201456, 77426676892, 903251324476, 10629495065550, 126032922655030, 1504194199010435, 18056321542477095, 217859030049153565, 2640609137351540510, 32137554969392230950, 392580762083089376630

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A025227, A381787, A381937.

Cf. A002293, A346647, A365184.

a(n) = sum(k=0, n, binomial(5*k+1, k)*binomial(k+1, n-k)/(5*k+1));

a(n) = Sum_{k=0..n} binomial(5*k+1,k) * binomial(k+1,n-k)/(5*k+1).

a(n) = A365184(n) + A365184(n-1).

A176698 A symmetrical sum triangle sequence:a(n)=vector(a(n-1)).Reverse(vector(a(n-1));a(0)=1;a(1)=2;t(n,m)=2+a(n)-a(m)-a(n-m).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 8, 8, 1, 1, 28, 34, 28, 1, 1, 104, 130, 130, 104, 1, 1, 400, 502, 522, 502, 400, 1, 1, 1584, 1982, 2078, 2078, 1982, 1584, 1, 1, 6416, 7998, 8390, 8466, 8390, 7998, 6416, 1, 1, 26464, 32878, 34454, 34826, 34826, 34454, 32878, 26464, 1, 1

Offset: 0

Roger L. Bagula, Apr 24 2010

Row sums are:

{1, 2, 4, 18, 92, 470, 2328, 11290, 54076, 257246, 1219296,...}.

			{1},
{1, 1},
{1, 2, 1},
{1, 8, 8, 1},
{1, 28, 34, 28, 1},
{1, 104, 130, 130, 104, 1},
{1, 400, 502, 522, 502, 400, 1},
{1, 1584, 1982, 2078, 2078, 1982, 1584, 1},
{1, 6416, 7998, 8390, 8466, 8390, 7998, 6416, 1},
{1, 26464, 32878, 34454, 34826, 34826, 34454, 32878, 26464, 1},
{1, 110784, 137246, 143654, 145210, 145506, 145210, 143654, 137246, 110784, 1}

Cf. A025227

a[0] := 1; a[1] := 2;
a[n_] := a[n] = Table[a[i], {i, 0, n - 1}].Table[a[n - 1 - i], {i, 0, n - 1}];
t[n_, m_] := 2 + (-a[m] - a[n - m] + a[n]);
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]

a(n)=vector(a(n-1)).Reverse(vector(a(n-1));

a(0)=1;a(1)=2;\q t(n,m)=2+a(n)-a(m)-a(n-m)

A200756 Triangle T(n,k) = coefficient of x^n in expansion of ((1 -sqrt(1 - 4x - 4x^2))/2)^k.

Original entry on oeis.org

1, 2, 1, 4, 4, 1, 12, 12, 6, 1, 40, 40, 24, 8, 1, 144, 144, 92, 40, 10, 1, 544, 544, 360, 176, 60, 12, 1, 2128, 2128, 1440, 752, 300, 84, 14, 1, 8544, 8544, 5872, 3200, 1400, 472, 112, 16, 1, 35008, 35008, 24336, 13664, 6352, 2400, 700, 144, 18, 1

Offset: 1

Vladimir Kruchinin, Nov 22 2011

Triangle T(n,k) =
1. Riordan Array (1, (1 - sqrt(1 - 4*x - 4*x^2))/2) without first column.
2. Riordan Array ((1 - sqrt(1 - 4*x - 4*x^2))/(2*x), (1 - sqrt(1 - 4*x - 4*x^2))/2) numbering triangle (0,0).
The array factorizes in the Bell subgroup of the Riordan group as (1 + x, x*(1 + x)) * (c(x), x*c(x)) = A030528 * A033184, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108. - Peter Bala, Dec 11 2015

			    1,
    2,   1,
    4,   4,   1,
   12,  12,   6,   1,
   40,  40,  24,   8,  1,
  144, 144,  92,  40, 10,  1,
  544, 544, 360, 176, 60, 12, 1

Cf. A025227 (column 1), A000108, A030528, A033184.

Table[k Sum[Binomial[i, n - i] Binomial[-k + 2 i - 1, i - 1]/i, {i, k, n}], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Dec 11 2015 *)

T(n,k):=k*(sum((binomial(i,n-i)*binomial(-k+2*i-1,i-1))/i,i,k,n));

tabl(nn) = {for (n=1, nn, for(k=1, n, print1(k*sum(i=k, n, binomial(i,n-i)*binomial(-k+2*i-1,i-1)/i),", ",);); print();); };
tabl(10); \\ Indranil Ghosh, Mar 04 2017

T(n,k) = k*( Sum_{i = k..n} binomial(i,n-i)*binomial(-k+2*i-1,i-1)/i );

cp's OEIS Frontend

A379173 G.f. A(x) satisfies A(x) = (1 + x)/(1 - x*A(x))^2.

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A052727 Expansion of e.g.f. 1/2-1/2*(1-4*x-4*x^2)^(1/2).

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A068765 Generalized Catalan numbers 3*x*A(x)^2 -A(x) +1 -2*x = 0.

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A176703 Coefficients of a recursive polynomial based on Chaitin's S expressions: a(0)=1; a(1)=x; a(2)=1; a(n)=vector(a(n-1)).reverse(a(n-1)).

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A290147 Expansion of (1-sqrt(1-8*x-8*x^2))/(4*x).

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A337991 Triangle read by rows: T(n,m) = Sum_{i=1..n} C(n,i-m)*C(n+m-i,i-1)*C(n+m-i,m)/n, with T(0,0)=1.

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A381937 G.f. A(x) satisfies A(x) = (1 + x) * B(x*A(x)), where B(x) is the g.f. of A001764.

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A381940 G.f. A(x) satisfies A(x) = (1 + x) * B(x*A(x)), where B(x) is the g.f. of A002293.

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A052727 Expansion of e.g.f. 1/2-1/2(1-4x-4*x^2)^(1/2).

A068765 Generalized Catalan numbers 3xA(x)^2 -A(x) +1 -2*x = 0.

A290147 Expansion of (1-sqrt(1-8x-8x^2))/(4*x).

A337991 Triangle read by rows: T(n,m) = Sum_{i=1..n} C(n,i-m)C(n+m-i,i-1)C(n+m-i,m)/n, with T(0,0)=1.

A200756 Triangle T(n,k) = coefficient of x^n in expansion of ((1 -sqrt(1 - 4x - 4x^2))/2)^k.