A107267 -id:A107267 - cp's OEIS frontend

A107268 Sums of antidiagonals of A107267.

1, 1, 2, 5, 14, 46, 173, 733, 3436, 17572, 96997, 573268, 3604687, 23990345, 168254444, 1238889493, 9546211068, 76761000444, 642524333589, 5586361188966, 50351455288661, 469653513479395, 4526242614854118, 45005754238016688

Offset: 0

Paul Barry, May 15 2005

Alois P. Heinz, Table of n, a(n) for n = 0..250

a(n) = Sum_{k=0..n} Sum_{j=0..n-k} 1/(j+1) C(j+1,n-k-j+1) C(n-k,j) k^j.

a(n) = Sum_{k=0..n} Sum_{j=0..k} 1/(j+1) C(j+1,k-j+1) C(k,j) (n-k)^j.

A107269 Diagonal sums of A107267, viewed as a number triangle.

Original entry on oeis.org

1, 0, 1, 1, 3, 6, 16, 44, 136, 457, 1669, 6547, 27334, 120465, 557094, 2692528, 13564940, 71102869, 387132791, 2185813423, 12775466408, 77154694819, 480630407368, 3083443893896, 20344187534051, 137884769704754

Offset: 0

Paul Barry, May 15 2005

a(n)=sum{k=0..n, sum{j=0..n-2k, (1/(j+1))C(j+1, n-2k-j+1)C(n-2k, j)k^j}}

A107264 Expansion of (1 - 3x - sqrt((1-3x)^2 - 43x^2))/(23x^2).

Original entry on oeis.org

1, 3, 12, 54, 261, 1323, 6939, 37341, 205011, 1143801, 6466230, 36960300, 213243435, 1240219269, 7263473148, 42799541886, 253556163243, 1509356586897, 9023497273548, 54154973176074, 326154592965879, 1970575690572297

Offset: 0

Paul Barry, May 15 2005

Series reversion of x/(1+3x+3x^2). Transform of 3^n under the matrix A107131. A row of A107267.
Counts colored Motzkin paths, where H(1,0) and U(1,1) each have 3 colors and D(1,-1) one color. - Paul Barry, May 18 2005
Number of Motzkin paths of length n in which both the "up" and the "level" steps come in three colors. - Paul Barry, May 18 2005
Third binomial transform of 1,0,3,0,18,0,... or 3^n*C(n) (A005159) with interpolated zeros. - Paul Barry, May 24 2005
As a continued fraction, the g.f. is 1/(1-3*x-3*x^2/(1-3*x-3*x^2/(1-3*x-3*x^2/(1-3*x-3*x^2/(.... - Paul Barry, Dec 02 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.
N. Gabriel, K. Peske, L. Pudwell, and S. Tay, Pattern Avoidance in Ternary Trees, J. Int. Seq. 15 (2012) # 12.1.5.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
L. Pudwell, Pattern avoidance in trees, (slides from a talk, mentions many sequences), 2012. - From _N. J. A. Sloane_, Jan 03 2013

CoefficientList[Series[(1-3*x-Sqrt[1-6*x-3*x^2])/(6*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

G.f.: (1 - 3x - sqrt(1-6x-3x^2))/(6x^2);
a(n) = Sum_{k=0..n} (1/(k+1))*C(k+1, n-k+1)*C(n, k)3^k.
a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)*C(k)*3^(n-k). - Paul Barry, May 18 2005
E.g.f.: exp(3x)*Bessel_I(1, sqrt(3)*2*x)/(sqrt(3)*x). - Paul Barry, May 24 2005
a(n) = (1/Pi)*Integral_{x=3-2*sqrt(3)..3+2*sqrt(3)} x^n*sqrt(-x^2 + 6*x + 3)/6. - Paul Barry, Sep 16 2006
a(n) = A156016(n+1)/3. - Philippe Deléham, Feb 04 2009
D-finite with recurrence: (n+2)*a(n) = 3*(2*n+1)*a(n-1) + 3*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ (5+3*sqrt(3))*(3+2*sqrt(3))^n/(2*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
G.f.: Let F(x) be the g.f. of A348189 with offset 1, then F(x) = x + 2*x^2*F(x)^2*A(x*F(x)). - Alexander Burstein, Feb 14 2022

A292716 a(n) = [x^n] 1/(1 - nx - nx^2/(1 - nx - nx^2/(1 - nx - nx^2/(1 - nx - nx^2/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 1, 6, 54, 672, 10625, 203256, 4554697, 116842496, 3373056027, 108134200000, 3809118341028, 146170521796608, 6066719073261639, 270692733123460480, 12917478278285156250, 656311833287586742272, 35364920064570086779227, 2014028255250518880457728, 120852950097737555898105210

Offset: 0

Ilya Gutkovskiy, Sep 21 2017

nonn

Also coefficient of x^n in the expansion of 1/(n+1) * (1 + n*x + n*x^2)^(n+1). - Seiichi Manyama, May 06 2019

Alois P. Heinz, Table of n, a(n) for n = 0..381

Main diagonal of A107267.

Cf. A001006, A247496.

Table[SeriesCoefficient[1/(1 - n x + ContinuedFractionK[-n x^2, 1 - n x, {i, 1, n}]), {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[2/(1 - n x + Sqrt[1 + n x ((n - 4) x - 2)]), {x, 0, n}], {n, 0, 19}]
Table[n! SeriesCoefficient[E^(n x) Hypergeometric0F1Regularized[2, n x^2], {x, 0, n}], {n, 0, 19}]
Flatten[{1, Table[Sum[Binomial[k+1, n-k+1] * Binomial[n, k] * n^k / (k+1), {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 05 2019 *)

{a(n) = polcoef((1+n*x+n*x^2)^(n+1)/(n+1), n)} \\ Seiichi Manyama, May 06 2019

a(n) = [x^n] 2/(1 - n*x + sqrt(1 + n*x*((n - 4)*x - 2))).
a(n) = n! * [x^n] exp(n*x)*BesselI(1,2*sqrt(n)*x)/(sqrt(n)*x), for n > 0.
a(n) = A107267(n,n).
a(n) ~ exp(2*sqrt(n) - 2) * n^(n - 3/4) / (2*sqrt(Pi)). - Vaclav Kotesovec, May 05 2019

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).

A307910 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2kx + k(k-4)x^2).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 8, 7, 0, 1, 4, 15, 32, 19, 0, 1, 5, 24, 81, 136, 51, 0, 1, 6, 35, 160, 459, 592, 141, 0, 1, 7, 48, 275, 1120, 2673, 2624, 393, 0, 1, 8, 63, 432, 2275, 8064, 15849, 11776, 1107, 0, 1, 9, 80, 637, 4104, 19375, 59136, 95175, 53344, 3139, 0

Offset: 0

Seiichi Manyama, May 05 2019

			Square array begins:
   1,   1,     1,     1,      1,       1,       1, ...
   0,   1,     2,     3,      4,       5,       6, ...
   0,   3,     8,    15,     24,      35,      48, ...
   0,   7,    32,    81,    160,     275,     432, ...
   0,  19,   136,   459,   1120,    2275,    4104, ...
   0,  51,   592,  2673,   8064,   19375,   40176, ...
   0, 141,  2624, 15849,  59136,  168125,  400896, ...
   0, 393, 11776, 95175, 439296, 1478125, 4053888, ...

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

Columns k=0..4 give A000007, A002426, A006139, A122868, A059304.
Main diagonal gives A092366.
Cf. A107267, A292627, A307819, A307847, A307855, A307883.

A[n_, k_] := k^n Hypergeometric2F1[(1-n)/2, -n/2, 1, 4/k]; A[0, ] = 1; A[, 0] = 0; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 07 2019 *)

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

A107265 Expansion of (1-5x-sqrt((1-5x)^2-45x^2))/(25x^2).

Original entry on oeis.org

1, 5, 30, 200, 1425, 10625, 81875, 646875, 5211875, 42659375, 353725000, 2965031250, 25083859375, 213894609375, 1836516718750, 15863968750000, 137767560546875, 1202116083984375, 10534061644531250, 92664360625000000, 817975366904296875, 7243402948779296875

Offset: 0

Paul Barry, May 15 2005

Series reversion of x/(1+5x+5x^2). Transform of 5^n under the matrix A107131. A row of A107267.

Counts colored Motzkin paths, where H(1,0) and U(1,1) each have 5 colors and D(1,-1) one color. - Paul Barry, May 16 2005

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

CoefficientList[Series[(1-5*x-Sqrt[1-10*x+5*x^2])/(10*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

x='x+O('x^66); Vec((1-5*x-sqrt(1-10*x+5*x^2))/(10*x^2)) \\ Joerg Arndt, May 15 2013

G.f.: (1-5*x-sqrt(1-10*x+5*x^2))/(10*x^2).
a(n) = Sum_{k=0..n} (1/(k+1)) * C(k+1,n-k+1) * C(n, k) * 5^k.
E.g.f.: a(n) = n!* [x^n] exp(5*x)*BesselI(1,2*sqrt(5)*x) /(sqrt(5)*x). -Peter Luschny, Aug 25 2012
D-finite with recurrence: (n+2)*a(n) = 5*(2*n+1)*a(n-1) - 5*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ sqrt(38+17*sqrt(5))*(5+2*sqrt(5))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
G.f.: 1/(1 - 5*x - 5*x^2/(1 - 5*x - 5*x^2/(1 - 5*x - 5*x^2/(1 - 5*x - 5*x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Sep 21 2017

A107266 Expansion of (1-6x-sqrt((1-6x)^2-46x^2))/(26x^2).

Original entry on oeis.org

1, 6, 42, 324, 2664, 22896, 203256, 1849392, 17156448, 161663040, 1543053888, 14887836288, 144963737856, 1422685140480, 14058304458624, 139754913276672, 1396721001457152, 14025182471414784, 141432971217841152, 1431708373864249344, 14543342842406252544, 148198801896234491904

Offset: 0

Paul Barry, May 15 2005

Series reversion of x/(1+6x+6x^2). Transform of 6^n under the matrix A107131. A row of A107267.

Counts colored Motzkin paths, where H(1,0) and U(1,1) each have 6 colors and D(1,-1) one color. - Paul Barry, May 16 2005

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

CoefficientList[Series[(1-6*x-Sqrt[1-12*x+12*x^2])/(12*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

x='x+O('x^66); Vec((1-6*x-sqrt(1-12*x+12*x^2))/(12*x^2)) \\ Joerg Arndt, May 15 2013

G.f.: (1-6*x-sqrt(1-12*x+12*x^2))/(12*x^2).
a(n) = Sum_{k=0..n} 1/(k+1) * C(k+1,n-k+1) * C(n,k) * 6^k.
E.g.f.: a(n) = n! * [x^n] exp(6*x)*BesselI(1, 2*sqrt(6)*x)/(sqrt(6)*x). -Peter Luschny, Aug 25 2012
D-finite with recurrence: (n+2)*a(n) = 6*(2*n+1)*a(n-1) - 12*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ sqrt(44+18*sqrt(6))*(6+2*sqrt(6))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
G.f.: 1/(1 - 6*x - 6*x^2/(1 - 6*x - 6*x^2/(1 - 6*x - 6*x^2/(1 - 6*x - 6*x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Sep 21 2017

A307968 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 + kx + sqrt(1 + 2k*x + k(k+4)x^2)).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 2, 2, 0, 1, -4, 6, 4, -3, 0, 1, -5, 12, 0, -24, -1, 0, 1, -6, 20, -16, -63, 48, 11, 0, 1, -7, 30, -50, -96, 297, 24, -15, 0, 1, -8, 42, -108, -75, 896, -621, -464, -13, 0, 1, -9, 56, -196, 72, 1875, -3904, -1053, 1376, 77, 0

Offset: 0

Seiichi Manyama, May 08 2019

			Square array begins:
   1,   1,    1,     1,     1,      1,      1, ...
   0,  -1,   -2,    -3,    -4,     -5,     -6, ...
   0,   0,    2,     6,    12,     20,     30, ...
   0,   2,    4,     0,   -16,    -50,   -108, ...
   0,  -3,  -24,   -63,   -96,    -75,     72, ...
   0,  -1,   48,   297,   896,   1875,   3024, ...
   0,  11,   24,  -621, -3904, -13125, -32184, ...
   0, -15, -464, -1053,  6912,  53125, 200880, ...

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

Columns k=0..2 give A000007, A007440(n+1), A307969.
Main diagonal gives A307946.
Cf. A107267, A307819.

T[n_, k_] := Sum[If[k == n-j == 0, 1, (-k)^(n-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 - k*x - k*x^2)^(n+1).
A(n,k) = Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n,j) * binomial(n-j,j)/(j+1) = Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n,2*j) * A000108(j).
(n+2) * A(n,k) = -k * (2*n+1) * A(n-1,k) - k * (k+4) * (n-1) * A(n-2,k).

cp's OEIS Frontend

A107268 Sums of antidiagonals of A107267.

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A107269 Diagonal sums of A107267, viewed as a number triangle.

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A107264 Expansion of (1 - 3*x - sqrt((1-3*x)^2 - 4*3*x^2))/(2*3*x^2).

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A292716 a(n) = [x^n] 1/(1 - n*x - n*x^2/(1 - n*x - n*x^2/(1 - n*x - n*x^2/(1 - n*x - n*x^2/(1 - ...))))), a continued fraction.

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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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A307910 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*k*x + k*(k-4)*x^2).

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A107265 Expansion of (1-5*x-sqrt((1-5*x)^2-4*5*x^2))/(2*5*x^2).

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A107266 Expansion of (1-6*x-sqrt((1-6*x)^2-4*6*x^2))/(2*6*x^2).

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A307968 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 + k*x + sqrt(1 + 2*k*x + k*(k+4)*x^2)).

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A107264 Expansion of (1 - 3x - sqrt((1-3x)^2 - 43x^2))/(23x^2).

A292716 a(n) = [x^n] 1/(1 - nx - nx^2/(1 - nx - nx^2/(1 - nx - nx^2/(1 - nx - nx^2/(1 - ...))))), a continued fraction.

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)).

A307910 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2kx + k(k-4)x^2).

A107265 Expansion of (1-5x-sqrt((1-5x)^2-45x^2))/(25x^2).

A107266 Expansion of (1-6x-sqrt((1-6x)^2-46x^2))/(26x^2).

A307968 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 + kx + sqrt(1 + 2k*x + k(k+4)x^2)).