A011270 -id:A011270 - cp's OEIS frontend

A365150 G.f. satisfies A(x) = 1 + xA(x)^2 / (1 - xA(x))^3.

1, 1, 5, 26, 150, 925, 5967, 39772, 271758, 1893431, 13400897, 96078789, 696333585, 5093266409, 37549674939, 278739057687, 2081637677823, 15628794649931, 117897848681271, 893167062280029, 6792410218680749, 51835002735642287, 396821349652564273

Offset: 0

Seiichi Manyama, Aug 23 2023

nonn

Cf. A001003, A011270.
Cf. A365151, A365152.
Cf. A052529.

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

If g.f. satisfies A(x) = ( 1 + x*A(x)^2 / (1 - x*A(x))^s )^t, then a(n) = Sum_{k=0..n} binomial(t*(n+k+1),k) * binomial(n+(s-1)*k-1,n-k)/(n+k+1).

G.f.: (1/x) * Series_Reversion( x*(1 - x/(1 - x)^3) ). - Seiichi Manyama, Sep 24 2024

A193589 Augmentation of the Fibonacci triangle A193588. See Comments.

Original entry on oeis.org

1, 1, 2, 1, 4, 7, 1, 6, 18, 31, 1, 8, 33, 90, 154, 1, 10, 52, 185, 481, 820, 1, 12, 75, 324, 1065, 2690, 4575, 1, 14, 102, 515, 2006, 6276, 15547, 26398, 1, 16, 133, 766, 3420, 12468, 37711, 92124, 156233, 1, 18, 168, 1085, 5439, 22412, 78030, 230277

Offset: 0

Clark Kimberling, Jul 31 2011

For an introduction to the unary operation augmentation as applied to triangular arrays or sequences of polynomials, see A193091.
Regarding A193589, if the triangle is written as (w(n,k)), then w(n,n)=A007863(n); w(n,n-1)=A011270; and
(col 3)=A033537.

			First 5 rows of A193588:
1
1....2
1....2....3
1....2....3....5
1....2....3....5....8
First 5 rows of A193589:
1
1....2
1....4....7
1....6....18...31
1....8....33...90...154

Cf. A193091, A193588.

p[n_, k_] := Fibonacci[k + 2]
Table[p[n, k], {n, 0, 5}, {k, 0, n}]  (* A193588 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]]  (* A193589 *)
Flatten[Table[v[n], {n, 0, 8}]]

A367232 G.f. satisfies A(x) = 1 + xA(x)^3 / (1 - xA(x))^2.

Original entry on oeis.org

1, 1, 5, 29, 189, 1325, 9757, 74429, 583037, 4662653, 37911037, 312457469, 2604534269, 21919435517, 185992729085, 1589480795133, 13668519794685, 118188894992381, 1026965424910333, 8962634482450429, 78528344593006589, 690502653622083581

Offset: 0

Seiichi Manyama, Nov 11 2023

nonn

Cf. A000108, A011270, A109081.

a(n, s=2, t=3, u=1) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+1));

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

A011272 Hybrid binary rooted trees with n nodes whose root is labeled by "a".

Original entry on oeis.org

0, 1, 3, 13, 64, 339, 1885, 10851, 64109, 386510, 2368354, 14706331, 92337618, 585239903, 3739309053, 24059542845, 155756019048, 1013801283133, 6630587014935, 43553555324502

Offset: 0

Jean Pallo (pallo(AT)u-bourgogne.fr)

nonn

Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
J. M. Pallo, On the listing and random generation of hybrid binary trees, International Journal of Computer Mathematics, 50 (1994) 135-145.
Index entries for sequences related to rooted trees

a(n) = A007863(n) - A011270(n).

A367234 G.f. satisfies A(x) = 1 + xA(x)^2 / (1 - xA(x))^4.

Original entry on oeis.org

1, 1, 6, 35, 226, 1561, 11276, 84150, 643730, 5021038, 39781858, 319282210, 2590312872, 21208628405, 175024439504, 1454329099044, 12157356271998, 102170610282040, 862721635191860, 7315768816166027, 62274763166575410, 531950072655682896, 4558282056420235664

Offset: 0

Seiichi Manyama, Nov 11 2023

nonn

Index entries for reversions of series

Cf. A321798, A365114, A367235.
Cf. A001003, A011270, A365150.
Cf. A361306, A378567.

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

If g.f. satisfies A(x) = 1 + x*A(x)^t / (1 - x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(n+(s-1)*k-1,n-k) / (t*k+u*(n-k)+1).
From Seiichi Manyama, Dec 01 2024: (Start)
G.f.: exp( Sum_{k>=1} A378567(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] 1/(1 - x/(1 - x)^4)^(n+1).
G.f.: (1/x) * Series_Reversion( x*(1 - x/(1 - x)^4) ). (End)

A378565 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(n+k-1,n-k).

Original entry on oeis.org

1, 1, 7, 43, 271, 1746, 11425, 75615, 504799, 3392953, 22930282, 155664356, 1060710457, 7250779238, 49700101101, 341474150583, 2351032782783, 16216401440106, 112035931072915, 775163096510445, 5370301986029066, 37249469056575504, 258648802856972348

Offset: 0

Seiichi Manyama, Dec 01 2024

nonn

Cf. A002002, A378566, A378567.

Cf. A011270, A362087.

Table[Sum[Binomial[n+k-1, k] * Binomial[n+k-1, 2*k-1], {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Dec 01 2024 *)

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

a(n) = [x^n] 1/(1 - x/(1 - x)^2)^n.

a(n) ~ (525 - 32*210^(2/3)/(157*sqrt(105) - 1575)^(1/3) + 4*(210*(157*sqrt(105) - 1575))^(1/3))^(1/6) * ((36 + (1208682 - 28350*sqrt(105))^(1/3)/3 + (6*(7461 + 175*sqrt(105)))^(1/3))^n / (2^(2/3) * 7^(1/3) * sqrt(Pi*n) * 3^(n + 1/6) * 5^(n + 1/3))). - Vaclav Kotesovec, Dec 01 2024

A011274 Triangle of numbers of hybrid rooted trees (divided by Fibonacci numbers).

Original entry on oeis.org

1, 2, 1, 7, 4, 1, 31, 18, 6, 1, 154, 90, 33, 8, 1, 820, 481, 185, 52, 10, 1, 4575, 2690, 1065, 324, 75, 12, 1, 26398, 15547, 6276, 2006, 515, 102, 14, 1, 156233, 92124, 37711, 12468, 3420, 766, 133, 16, 1, 943174, 556664, 230277, 78030, 22412, 5439, 1085, 168, 18, 1

Offset: 1

Jean Pallo (pallo(AT)u-bourgogne.fr)

Triangle T(n,k) = [x^(n-k)] A(x)^k where A(x) is the o.g.f. of A007863. - Vladimir Kruchinin, Mar 17 2011

Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A007863. - Philippe Deléham, Feb 03 2014

			     1
     2    1
     7    4    1
    31   18    6   1
   154   90   33   8  1
   820  481  185  52 10  1
  4575 2690 1065 324 75 12 1
Production matrix is:
   2   1
   3   2   1
   5   3   2   1
   8   5   3   2   1
  13   8   5   3   2   1
  21  13   8   5   3   2   1
  34  21  13   8   5   3   2   1
  55  34  21  13   8   5   3   2   1
  89  55  34  21  13   8   5   3   2   1
  ... - _Philippe Deléham_, Feb 03 2014

Vincenzo Librandi, Rows n = 1..100, flattened
Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
J. M. Pallo, On the listing and random generation of hybrid binary trees, International Journal of Computer Mathematics, 50 (1994) 135-145.
Index entries for sequences related to rooted trees

Cf. A000045, A011270, A011272.

A011274 := proc(n,k) k/n*add( binomial(i+n-1,n-1)*binomial(i+n,n-k-i),i=0..n-k) ; end proc: # R. J. Mathar, Mar 21 2011

t[n_, k_] := k/n*Binomial[n, k]*HypergeometricPFQ[ {k-n, n, n+1}, {1/2 + k/2, 1+k/2}, -1/4]; Flatten[ Table[ t[n, k], {n, 1, 10}, {k, 1, n}]] (* Jean-François Alcover, Dec 02 2011, after Vladimir Kruchinin *)

A011274(n,k):= k/n*sum(binomial(i+n-1,n-1)*binomial(i+n,n-k-i), i,0,n-k); /* Vladimir Kruchinin, Mar 17 2011 */

T(n,k) = (k/n) *Sum_{i=0..n-k} binomial(i+n-1,n-1)*binomial(i+n,n-k-i). - Vladimir Kruchinin, Mar 17 2011
(r/(m*n+r))*T((m+1)*n+r,m*n+r) = Sum_{k=1..n} k*T((m+1)*n-k,m*n)*T(r+k,r)/n. - Vladimir Kruchinin, Mar 17 2011
T(n,m) = (m/n)*Sum_{k=1..n-m+1} k*A007863(k-1)*T(n-k,m-1), 1 < m <= n. - Vladimir Kruchinin, Mar 17 2011

A365148 G.f. satisfies A(x) = ( 1 + xA(x)^2 / (1 - xA(x))^2 )^2.

Original entry on oeis.org

1, 2, 13, 102, 898, 8484, 84061, 861918, 9068950, 97366812, 1062425010, 11747773372, 131350499044, 1482494173128, 16867912278237, 193273940978574, 2228186999313678, 25827663921909228, 300825086742672934, 3519001122784601524, 41325186203051759324

Offset: 0

Seiichi Manyama, Aug 23 2023

nonn

Cf. A011270, A365149.

Cf. A365120, A365133.

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

If g.f. satisfies A(x) = ( 1 + x*A(x)^2 / (1 - x*A(x))^s )^t, then a(n) = Sum_{k=0..n} binomial(t*(n+k+1),k) * binomial(n+(s-1)*k-1,n-k)/(n+k+1).

A365149 G.f. satisfies A(x) = ( 1 + xA(x)^2 / (1 - xA(x))^2 )^3.

Original entry on oeis.org

1, 3, 27, 301, 3780, 51030, 723170, 10611594, 159845946, 2457515235, 38406398016, 608330707740, 9744053489754, 157564967282709, 2568706865998272, 42173100349112852, 696692754641035014, 11572241797209975966, 193153224033985241217

Offset: 0

Seiichi Manyama, Aug 23 2023

nonn

Cf. A011270, A365148.

Cf. A365121, A365134.

a(n, s=2, t=3) = sum(k=0, n, binomial(t*(n+k+1), k)*binomial(n+(s-1)*k-1, n-k)/(n+k+1));

If g.f. satisfies A(x) = ( 1 + x*A(x)^2 / (1 - x*A(x))^s )^t, then a(n) = Sum_{k=0..n} binomial(t*(n+k+1),k) * binomial(n+(s-1)*k-1,n-k)/(n+k+1).

cp's OEIS Frontend

A365150 G.f. satisfies A(x) = 1 + x*A(x)^2 / (1 - x*A(x))^3.

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A193589 Augmentation of the Fibonacci triangle A193588. See Comments.

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A367232 G.f. satisfies A(x) = 1 + x*A(x)^3 / (1 - x*A(x))^2.

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A011272 Hybrid binary rooted trees with n nodes whose root is labeled by "a".

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A367234 G.f. satisfies A(x) = 1 + x*A(x)^2 / (1 - x*A(x))^4.

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A378565 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(n+k-1,n-k).

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A011274 Triangle of numbers of hybrid rooted trees (divided by Fibonacci numbers).

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A365148 G.f. satisfies A(x) = ( 1 + x*A(x)^2 / (1 - x*A(x))^2 )^2.

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A365149 G.f. satisfies A(x) = ( 1 + x*A(x)^2 / (1 - x*A(x))^2 )^3.

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A365150 G.f. satisfies A(x) = 1 + xA(x)^2 / (1 - xA(x))^3.

A367232 G.f. satisfies A(x) = 1 + xA(x)^3 / (1 - xA(x))^2.

A367234 G.f. satisfies A(x) = 1 + xA(x)^2 / (1 - xA(x))^4.

A365148 G.f. satisfies A(x) = ( 1 + xA(x)^2 / (1 - xA(x))^2 )^2.

A365149 G.f. satisfies A(x) = ( 1 + xA(x)^2 / (1 - xA(x))^2 )^3.