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 1-4 of 4 results.

A073153 Triangle of numbers relating two sequences A073155 and A073156.

Original entry on oeis.org

1, 1, 2, 4, 5, 9, 14, 18, 22, 36, 56, 70, 86, 100, 156, 237, 293, 349, 405, 461, 698, 1046, 1283, 1507, 1703, 1927, 2164, 3210, 4762, 5808, 6756, 7540, 8324, 9272, 10318, 15080, 22198, 26960, 31144, 34462, 37598, 40916, 45100, 49862, 72060, 105430, 127628
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2002

Keywords

Examples

			T(4,0) = T(3,3) + 2*T(2,2) + T(1,1) = 2 + 2*9 + 36 = 56.
T(5,2) = A073155(0)*A073155(5) + A073155(1)*A073155(4) + A073155(2)*A073155(3) = 1*237 + 1*56 + 4*14 = 349.
Triangle begins:
1;
1, 2;
4, 5, 9;
14, 18, 22, 36;
56, 70, 86, 100, 156;
237, 293, 349, 405, 461, 698;
1046, 1283, 1507, 1703, 1927, 2164, 3210; ...

Crossrefs

Cf. A073154, A073155, A073156, A073157, A073153.

Formula

Triangle {T(n, k), n >= 0, 0<=k<=n} defined by: T(0, 0)=1,
T(n, 0) = A073155(n), T(n, n) = A073156(n),
T(n, 0) = T(n-1, n-1) + 2*T(n-2, n-2) + T(n-3, n-3),
T(n, k) = Sum_{j=0..k} A073155(j) * A073155(n-j).

Extensions

Formulas simplified by Paul D. Hanna, Nov 26 2012
a(31) onward corrected by Sean A. Irvine, Nov 18 2024

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

A073155 Leftmost column sequence of triangle A073153.

Original entry on oeis.org

1, 1, 4, 14, 56, 237, 1046, 4762, 22198, 105430, 508384, 2482297, 12248416, 60980875, 305955356, 1545397464, 7852100294, 40105277640, 205798130604, 1060467961508, 5485199090812, 28469067353686, 148220323891460
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2002

Keywords

Examples

			a(3)=a(0)*[a(2)+a(1)]+[a(1)+a(0)]*[a(1)+a(0)]+[a(2)+a(1)]*a(0) =1*[4+1] + [1+1]*[1+1] + [4+1]*1 = 5 + 2*2 + 5 = 14.

Links

Crossrefs

Cf. A073153, A073156, A073157.
Cf. A052709, A360076, A360082, A360083.

Formula

Convolution of sequence formed from sum of adjacent terms yields the original sequence without the first term:
a(n+1) = Sum_{k=0..n} [a(k) + a(k-1)] * [a(n-k) + a(n-k-1)], where a(-1)=0.
G.f.: 1/2*(1-(1-4*x*(1+x)^2)^(1/2))/x/(1+x)^2. - Vladeta Jovovic, Oct 10 2003
a(n) = Sum_{k=0..n} C(2k,n-k)*C(k). - Paul Barry, Jul 09 2006
Conjecture: (n+1)*a(n) + (-3*n+4)*a(n-1) + 2*(-6*n+7)*a(n-2) + 2*(-6*n+11)*a(n-3) + 2*(-2*n+5)*a(n-4)=0. - R. J. Mathar, Nov 26 2012
G.f. A(x) satisfies: A(x) = 1 + x * ((1 + x) * A(x))^2. - Ilya Gutkovskiy, Jul 10 2020

Extensions

More terms from Vladeta Jovovic, Oct 10 2003

A073154 Triangle of numbers relating two sequences (A073157 and A073155).

Original entry on oeis.org

1, 2, 4, 5, 9, 14, 18, 28, 38, 56, 70, 106, 131, 167, 237, 293, 433, 523, 613, 753, 1046, 1283, 1869, 2219, 2543, 2893, 3479, 4762, 5808, 8374, 9839, 11099, 12359, 13824, 16390, 22198
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2002

Keywords

Examples

			a(4,0)=a(3,3)+a(2,2)=56+14=70.
a(5,2)=A073157(0)*A073157(5)+A073157(1)*A073157(4)+A073157(2)*A073157(3)= 1*293+2*70+5*18=523.
Rows:
  {1};
  {2,4};
  {5,9,14};
  {18,28,38,56};
  {70,106,131,167,237};
  {293,433,523,613,753,1046};
  {1283,1869,2219,2543,2893,3479,4762};
  ...

Crossrefs

Cf. A073153, A073155, A073156, A073157, A073152.

Formula

Triangle {a(n, k), n >= 0, 0<=k<=n} defined by: a(0, 0)=1, a(n, 0)=A073157(n), a(n, n)=A073155(n+1), a(n, 0)=a(n-1, n-1) + a(n-2, n-2), a(n, k)=sum{j=0..k} A073157(j) * A073157(n-j).
G.f.: Sum_{n>=0, 0<=k<=n} a(n, k) x^n y^k = A(x*y)(A(x) - y A(x*y))/(1 - y) where A(x) = (1 - (1 - 4 x (1 + x)^2)^(1/2))/(2 x (1 + x)) is the o.g.f. for A073157. - David Callan, Aug 16 2006
Showing 1-4 of 4 results.

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