cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A073157 Number of Schroeder n-paths containing no FFs.

Original entry on oeis.org

1, 2, 5, 18, 70, 293, 1283, 5808, 26960, 127628, 613814, 2990681, 14730713, 73229291, 366936231, 1851352820, 9397497758, 47957377934, 245903408244, 1266266092112, 6545667052320, 33954266444498, 176689391245146
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2002

Keywords

Comments

Number of Schroeder n-paths containing no FFs. A Schroeder n-path (A006318) consists of steps U=(1,1),F=(2,0),D=(1,-1) starting at (0,0), ending at (2n,0), and never going below the x-axis. Example: a(2)=5 counts UFD, UUDD, UDF, FUD, UDUD. - David Callan, Aug 23 2011

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 18*x^3 + 70*x^4 + 293*x^5 + 1283*x^6 + ...

Links

Crossrefs

Leftmost column of triangle A073154 (was previous name).
Cf. A000108, A073155, A073156, A073153, A143330, A198953, A215715, A216359, A364335, A364371.

Programs

  • GAP
    List([0..25],n->Sum([0..n],i->Binomial(2*i+1,i)*Binomial(2*i+1,n-i)/(2*i+1))); # Muniru A Asiru, Oct 11 2018
  • Maple
    a:=n->add(binomial(2*i+1,i)*binomial(2*i+1,n-i)/(2*i+1),i=0..n): seq(a(n),n=0..25); # Muniru A Asiru, Oct 11 2018
  • Mathematica
    Table[Sum[Binomial[2*i + 1, i]*Binomial[2*i + 1, n - i]/(2*i + 1), {i, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Oct 11 2018 *)
  • Maxima
    a(n):=sum((sum((binomial(2*k+2,j-k)*binomial(2*k,k)/(k+1)),k,0,j))*(-1)^(n-j),j,0,n); /* Vladimir Kruchinin, Mar 13 2016 */
  • PARI
    {a(n)=local(A=1); for(i=0,n-1,A=(1+x)*(1+x*(A+x*O(x^n))^2));polcoeff(A,n)} /* Paul D. Hanna, Mar 03 2008 */

Formula

A073155(n+1) = Sum_{k=0..n} a(k)*a(n-k), that is, convolution yields sequence A073155 minus the 0th term.
G.f.: A(x) = (1 - sqrt(1 - 4*x*(1+x)^2))/(2*x*(1+x)) satisfies A(x) = (1+x)*(1 + x*A(x)^2);
G.f.: A(x) = (1+x)*C(x*(1+x)^2) where C(x) is the Catalan g.f. of A000108. - Paul D. Hanna, Mar 03 2008
a(n) = Sum_{j=0..n}((Sum_{k=0..j}((binomial(2*k+2,j-k)*C(k))))*(-1)^(n-j)), where C(k) = A000108(k). - Vladimir Kruchinin, Mar 13 2016
a(n) = Sum_{i=0..n} C(2*i+1,i)*C(2*i+1,n-i)/(2*i+1). - Vladimir Kruchinin, Oct 11 2018
Recurrence: (n+1)*a(n) = 3*(n-1)*a(n-1) + 6*(2*n - 3)*a(n-2) + 6*(2*n - 5)*a(n-3) + 2*(2*n - 7)*a(n-4). - Vaclav Kotesovec, Oct 11 2018
From Peter Bala, Aug 25 2024: (Start)
(1/x) * series_reversion(x/A(x)) = 1 + 2*x + 9*x^2 + 56*x^3 + 400*x^4 + 3095*x^5 + 25240*x^6 + ... is the g.f. of A198953.
(1/x) * series_reversion(x*A(-x)) = 1 + 2*x + 3*x^2 + 8*x^3 + 25*x^4 + 83*x^5 + 289*x^6 + ... = G(x) + x, where G(x) = (1 - x^2 - sqrt(1 - 4*x - 2*x^2 + x^4))/(2*x) is the g.f. of A143330. (End)
Define a sequence operator R: {u(n): n >= 0} -> {v(n): n >= 0} by Sum_{n >= 0} v(n)*x^n = (1/x) * series_reversion(x/Sum_{n >= 0} u(n)*x^n). Then R({a(n)}) = A198953, R^2({a(n)}) = A215715 and R^3({a(n)}) = A364335. Cf. A216359. - Peter Bala, Sep 13 2024

Extensions

More terms from Paul D. Hanna, Mar 03 2008
New name using a comment from David Callan, Peter Luschny, Oct 14 2018

A384747 Triangle read by rows: T(n,k) is the number of rooted ordered trees with node weights summing to n, where the root has weight 0, non-root node weights are in {1,..,k}, and no nodes have the same weight as their parent node.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 5, 6, 0, 1, 11, 15, 16, 0, 1, 26, 39, 43, 44, 0, 1, 63, 110, 123, 127, 128, 0, 1, 153, 308, 358, 371, 375, 376, 0, 1, 376, 869, 1046, 1096, 1109, 1113, 1114, 0, 1, 931, 2499, 3098, 3278, 3328, 3341, 3345, 3346, 0, 1, 2317, 7238, 9283, 9904, 10084, 10134, 10147, 10151, 10152
Offset: 0

Views

Author

John Tyler Rascoe, Jun 09 2025

Keywords

Examples

			Triangle begins:
    k=0  1    2     3     4     5     6     7     8     9
 n=0 [1]
 n=1 [0, 1]
 n=2 [0, 1,   2]
 n=3 [0, 1,   5,    6]
 n=4 [0, 1,  11,   15,   16]
 n=5 [0, 1,  26,   39,   43,   44]
 n=6 [0, 1,  63,  110,  123,  127,  128]
 n=7 [0, 1, 153,  308,  358,  371,  375,  376]
 n=8 [0, 1, 376,  869, 1046, 1096, 1109, 1113, 1114]
 n=9 [0, 1, 931, 2499, 3098, 3278, 3328, 3341, 3345, 3346]
...
T(3,3) = 6 counts:
  o    o    o      o        o        __o__
  |    |    |     / \      / \      /  |  \
 (3)  (2)  (1)  (1) (2)  (2) (1)  (1) (1) (1)
       |    |
      (1)  (2)

Crossrefs

Cf. A051286 (column k=2), A382096 (column k=3), A384748 (main diagonal).
Cf. A000108, A002212, A143330, A384613, A384685.

Programs

  • PARI
    b(i,j,k,N) = {if(k>N,1, 1/( 1  - sum(u=1,j, if(u==i,0,x^u * b(u,j,k+1,N-u+1)))))}
Gx(k,N) = {my(x='x+O('x^(N+1))); Vec(1/(1 - sum(i=1,k, b(i,k,1,N)*x^i)))}
T(max_row) = { my( N = max_row+1, v = vector(N, i, if(i==1, 1, 0))~); for(k=1, N, v=matconcat([v, Gx(k,N)~])); vector(N, n, vector(n, k, v[n, k]))}
T(9)

Formula

T(n,k) = T(n,n) for k > n.

A171199 G.f. satisfies: A(x) = exp( Sum_{n>=1} [A(x)^n + A(x)^-n]*x^n/n ).

Original entry on oeis.org

1, 2, 3, 8, 25, 83, 289, 1041, 3847, 14504, 55569, 215727, 846761, 3354858, 13398965, 53888063, 218053915, 887107888, 3626373205, 14887942624, 61358959587, 253771944529, 1052917272543, 4381374717994, 18280470530047
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Comments

Same as A143330 after initial terms.

Examples

			G.f.: A(x) = 1 + 2*x + 3*x^2 + 8*x^3 + 25*x^4 + 83*x^5 + 289*x^6 +...
log(A(x)) = [A(x)+1/A(x)]*x + [A(x)^2+1/A(x)^2]*x^2/2 + [A(x)^3+1/A(x)^3]*x^3/3 +...

Crossrefs

Cf. A171190, A171191, A143330.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(sum(m=1,n,(A^m+A^-m+x*O(x^n))*x^m/m)));polcoeff(A,n)}
  • PARI
    {a(n)=polcoeff((1+x^2-sqrt((1-x^2)^2-4*x+x^2*O(x^n)))/(2*x), n)}

Formula

G.f.: A(x) = (1+x^2 - sqrt(1 - 4*x - 2*x^2 + x^4))/(2*x).
G.f. satisfies: 1 = (A(x) - x)*(1 - x*A(x)).
a(0) = 1, a(1) = 2; a(n) = a(n-1) + a(n-2) + Sum_{k=2..n-1} a(k) * a(n-k-1). - Ilya Gutkovskiy, Jul 20 2021

A384685 Triangle read by rows: T(n,k) is the number of rooted ordered trees with node weights summing to n, where the root has weight 0, all internal nodes have weight 1, and leaf nodes have weights in {1,...,k}.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 5, 8, 9, 0, 14, 25, 28, 29, 0, 42, 83, 95, 98, 99, 0, 132, 289, 337, 349, 352, 353, 0, 429, 1041, 1236, 1285, 1297, 1300, 1301, 0, 1430, 3847, 4652, 4854, 4903, 4915, 4918, 4919, 0, 4862, 14504, 17865, 18709, 18912, 18961, 18973, 18976, 18977
Offset: 0

Views

Author

John Tyler Rascoe, Jun 06 2025

Keywords

Examples

			Triangle begins:
    k=0     1     2     3     4     5     6     7      8
 n=0 [1]
 n=1 [0,    1]
 n=2 [0,    2,    3]
 n=3 [0,    5,    8,    9]
 n=4 [0,   14,   25,   28,   29]
 n=5 [0,   42,   83,   95,   98,   99]
 n=6 [0,  132,  289,  337,  349,  352,  353]
 n=7 [0,  429, 1041, 1236, 1285, 1297, 1300, 1301]
 n=8 [0, 1430, 3847, 4652, 4854, 4903, 4915, 4918, 4919]
...
T(2,2) = 3 counts:
  o    o      o
  |    |     / \
 (2)  (1)  (1) (1)
       |
      (1)

Crossrefs

Cf. (column k=1) A000108, A078481, A078482, A088218, (column k=2) A143330, A380761, A384613.

Programs

  • PARI
    b(k) = {(x^2-x^(k+1))/(1-x)}
P(N,k) = {my(x='x+O('x^N)); Vec((1-b(k)-sqrt((b(k)-1)^2-4*x))/(2*x))}
T(max_row) = { my( N = max_row+1, v = vector(N, i, if(i==1,1,0))~); for(k=1,N, v=matconcat([v,P(N+1,k)~])); vector(N,n, vector(n,k,v[n,k]))}

Formula

G.f. of column k is (1 - b(k,x) - sqrt((b(k,x) - 1)^2 - 4*x))/(2*x) where b(k,x) = (x^2 - x^(k + 1))/(1 - x).
T(n,k) = T(n,n) for k > n.

A384748 Number of rooted ordered trees with node weights summing to n, where the root has weight 0, non-root node weights are greater than 0, and no nodes have the same weight as their parent node.

Original entry on oeis.org

1, 1, 2, 6, 16, 44, 128, 376, 1114, 3346, 10152, 31028, 95474, 295532, 919446, 2873388, 9015812, 28390466, 89689586, 284173096, 902780060, 2875016084, 9176388532, 29349499212, 94050228650, 301918397716, 970815092346
Offset: 0

Views

Author

John Tyler Rascoe, Jun 09 2025

Keywords

Examples

			a(3) = 6 counts:
  o    o    o      o        o        __o__
  |    |    |     / \      / \      /  |  \
 (3)  (2)  (1)  (1) (2)  (2) (1)  (1) (1) (1)
       |    |
      (1)  (2)

Crossrefs

Cf. A000108, A002212, A143330, A384613, A384685, (main diagonal of A384747).

Programs

  • PARI
    b(i,j,k,N) = {if(k>N,1, 1/(1-sum(u=1,j, if(u==i,0,x^u*b(u,j,k+1,N-u+1)))))}
Dx(N) = {my(x='x+O('x^(N+1))); Vec(1/(1 - sum(i=1,N, b(i,N,1,N)*x^i)))}
Dx(10)

Extensions

a(14)-a(26) from David Radcliffe, Jun 10 2025

A307733 a(0) = a(1) = 1; a(n) = a(n-1) + a(n-2) + Sum_{k=0..n-1} a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 4, 14, 54, 220, 934, 4090, 18344, 83850, 389214, 1829736, 8693962, 41685714, 201442188, 980091814, 4797070022, 23603701828, 116688837886, 579312087802, 2887020896016, 14437318756818, 72424982972862, 364366674463824, 1837954750285458
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 05 2020

Keywords

Crossrefs

Cf. A002212, A004148, A006318, A085139, A128720, A143330, A171416, A175934, A245734.

Programs

  • Mathematica
    a[0] = a[1] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] + Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]
nmax = 24; CoefficientList[Series[(1 - x - x^2 - Sqrt[1 - 6 x + 3 x^2 + 2 x^3 + x^4])/(2 x), {x, 0, nmax}], x]

Formula

G.f. A(x) satisfies: A(x) = (1 - x + x*A(x)^2) / (1 - x - x^2).
G.f.: (1 - x - x^2 - sqrt(1 - 6*x + 3*x^2 + 2*x^3 + x^4)) / (2*x).

A364477 G.f. satisfies A(x) = 1 + x*A(x)^2 + x^2*A(x)^7.

Original entry on oeis.org

1, 1, 3, 14, 76, 448, 2791, 18078, 120516, 821435, 5698422, 40101623, 285583775, 2054272430, 14903954415, 108932920861, 801350333186, 5928653489398, 44084056075057, 329279673851792, 2469493161891742, 18588339309502760, 140383789476473354
Offset: 0

Views

Author

Seiichi Manyama, Jul 26 2023

Keywords

Crossrefs

Cf. A001002, A001764, A006605, A052709, A143330, A364473.
Cf. A364476.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(2*n+3*k, k)*binomial(2*n+2*k, n-2*k)/(n+4*k+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(2*n+3*k,k) * binomial(2*n+2*k,n-2*k) / (n+4*k+1).

A364473 G.f. satisfies A(x) = 1 + x*A(x)^2 + x^2*A(x)^6.

Original entry on oeis.org

1, 1, 3, 13, 65, 353, 2024, 12057, 73890, 462851, 2950261, 19073921, 124776881, 824409052, 5493384031, 36874564529, 249114808794, 1692489908494, 11556616157589, 79265016880139, 545860966841247, 3772800724433931, 26162662010039826, 181974370638420829
Offset: 0

Views

Author

Seiichi Manyama, Jul 26 2023

Keywords

Crossrefs

Cf. A001002, A001764, A006605, A052709, A143330, A364477.
Cf. A364472, A364474.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(2*n+2*k, k)*binomial(2*n+k, n-2*k)/(n+3*k+1));

Formula

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

A382096 Number of rooted ordered trees with node weights summing to n, where the root has weight 0, non-root node weights are in {1,2,3}, and no nodes have the same weight as their parent node.

Original entry on oeis.org

1, 1, 2, 6, 15, 39, 110, 308, 869, 2499, 7238, 21086, 61871, 182523, 540830, 1609238, 4805871, 14398559, 43264896, 130347450, 393650751, 1191441349, 3613345360, 10978726634, 33414836743, 101863289331, 310984519412, 950734751040, 2910319385881, 8919643999157, 27368321239074
Offset: 0

Views

Author

John Tyler Rascoe, Jun 08 2025

Keywords

Examples

			a(3) = 6 counts:
  o    o    o      o        o        __o__
  |    |    |     / \      / \      /  |  \
 (3)  (2)  (1)  (1) (2)  (2) (1)  (1) (1) (1)
       |    |
      (1)  (2)

Crossrefs

Cf. A000108, A002212, A143330, A384613, A384685, (column k=3 of A384747).

Programs

  • PARI
    b(i,j,k,N) = {if(k>N,1, 1/(1-sum(u=1,j, if(u==i,0,x^u*b(u,j,k+1,N-u+1)))))}
Gx(k,N) = {my(x='x+O('x^(N+1))); Vec(1/(1-sum(i=1,k, b(i,k,1,N)*x^i)))}
Gx(3,20)

Formula

G.f.: G(x) = 1/(1 - b_1(x)*x - b_2(x)*x^2 - b_3(x)*x^3) where b_1(x) = 1/(1 - b_2(x)*x^2 - b_3(x)*x^3), b_2(x) = 1/(1 - b_1(x)*x - b_3(x)*x^3), b_3(x) = 1/(1 - b_1(x)*x - b_2(x)*x^2).
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.