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.

A130524 Diagonal immediately below the main diagonal of square array A130523.

Original entry on oeis.org

1, 4, 18, 87, 442, 2332, 12677, 70605, 401172, 2317683, 13578615, 80502845, 482140580, 2912954129, 17733375207, 108676158775, 669914021414, 4151053001800, 25841001981211, 161534820531068, 1013566626969398, 6381398103680604, 40301852983776764, 255249505209864803, 1620819715569894894
Offset: 0

Views

Author

Paul D. Hanna, Jun 02 2007

Keywords

Crossrefs

Cf. A130523; diagonals: A007857, A130525; related: A000108, A001764.

Programs

  • PARI
    {a(n) = my(C,F,D); C=Ser(vector(n+1,r,binomial(2*r-2,r-1)/r)); F=Ser(vector(n+1,r,binomial(3*r-3,r-1)/(2*r-1))); D=1/(1-x*C*F-x*F^2); polcoef(C*D*F+x*O(x^n),n,x)}
for(n=0,25,print1(a(n),", "))

Formula

G.f.: A(x) = C(x)*D(x)*F(x), where D(x) = 1/(1 - x*C(x)*F(x) - x*F(x)^2) is the g.f. of the main diagonal (A007857), C(x) = 1 + x*C(x)^2 is the g.f. of Catalan numbers (A000108) and F(x) = 1 + x*F(x)^3 is the g.f. of ternary numbers (A001764).

Extensions

Edited and corrected by Paul D. Hanna, Jan 27 2025

A130525 Diagonal immediately above the main diagonal of square array A130523.

Original entry on oeis.org

1, 3, 13, 63, 324, 1733, 9541, 53725, 308085, 1793528, 10574165, 63018105, 379061652, 2298508911, 14035748542, 86240951745, 532812883413, 3307967729867, 20627845299471, 129141164822496, 811394148828087, 5114638998643903, 32336393838083539, 205000199138736499, 1302892014385402691
Offset: 0

Views

Author

Paul D. Hanna, Jun 02 2007

Keywords

Crossrefs

Cf. A130523; diagonals: A007857, A130524; related: A000108, A001764.

Programs

  • PARI
    {a(n) = my(C,F,D); C=Ser(vector(n+1,r,binomial(2*r-2,r-1)/r)); F=Ser(vector(n+1,r,binomial(3*r-3,r-1)/(2*r-1))); D=1/(1-x*C*F-x*F^2); polcoef(D*F+x*O(x^n),n,x)}
for(n=0,25,print1(a(n),", "))

Formula

G.f.: A(x) = D(x)*F(x), where D(x) = 1/(1 - x*C(x)*F(x) - x*F(x)^2) is the g.f. of the main diagonal (A007857), C(x) = 1 + x*C(x)^2 is the g.f. of Catalan numbers (A000108) and F(x) = 1 + x*F(x)^3 is the g.f. of ternary numbers (A001764).

Extensions

Edited and corrected by Paul D. Hanna, Jan 27 2025

A007857 Number of independent sets in rooted plane trees on n nodes.

Original entry on oeis.org

1, 2, 8, 37, 184, 959, 5172, 28641, 162008, 932503, 5445934, 32197334, 192357788, 1159603592, 7045356104, 43098733353, 265240985112, 1641100253735, 10202295895890, 63696629668980, 399216722146770, 2510833297584165
Offset: 1

Views

Author

Martin Klazar

Keywords

Comments

Equals the main diagonal of square array A130523. - Paul D. Hanna, Jun 06 2007
From Petros Hadjicostas, Aug 06 2020: (Start)
To prove R. J. Mathar's conjecture, let b(n) = A007226(n-1) = (2*/n)*binomial(3*(n-1), n-1) and c(n) = A000108(n-1) = binomial(2*(n-1), n-1)/n. Since a(n) = b(n) - c(n), it is enough to prove that each of the sequences (b(n): n >= 1) and (c(n): n >= 1) satisfies the same recurrence as (a(n): n >= 1).
For simplicity, denote the recurrence by f(n,0)*a(n) + f(n,1)*a(n-1) + f(n,2)*a(n-2) + f(n,3)*a(n-3) = 0. Let g(n) = 2*n*(2*n - 3)/(3*(3*n - 4)*(3*n - 5)) and h(n) = n/(2*(2*n - 3)). Then we can easily show that b(n-i) = b(n)* Product_{j=0..i-1} g(n-j) and c(n-i) = c(n)*Product_{j=0..i-1} h(n-j) for i >= 1.
Using a CAS (e.g. PARI), one can show that f(n,0) + f(n,1)*g(n) + f(n,2)*g(n)*g(n-1) + f(n,3)*g(n)*g(n-1)*g(n-2) = 0. Multiplying both sides by b(n), we get f(n,0)*b(n) + f(n,1)*b(n-1) + f(n,2)*b(n-2) + f(n,3)*b(n-3) = 0.
Again, using a CAS, one can show that f(n,0) + f(n,1)*h(n) + f(n,2)*h(n)*h(n-1) + f(n,3)*h(n)*h(n-1)*h(n-2) = 0. Multiplying both sides by c(n), we get f(n,0)*c(n) + f(n,1)*c(n-1) + f(n,2)*c(n-2) + f(n,3)*c(n-3) = 0. (End)

Links

Crossrefs

Cf. A000108, A001764, A007226, A130523.

Programs

Formula

a(n+1) = (2/(n+1))*C(3*n, n) - (1/(n+1))*C(2*n, n) = A007226(n) - A000108(n). - Paul Barry, Nov 05 2006
G.f.: A(x) = x/(1 - x*C(x)*F(x) - x*F(x)^2), where C(x) is g.f. of the Catalan numbers (A000108) (i.e., C(x) = 1 + x*C(x)^2) and F(x) is the g.f. of ternary numbers (A001764) (i.e., F(x) = 1 + x*F(x)^3). - Paul D. Hanna, Jun 06 2007
Conjecture: 2*n*(n - 1)*(2*n - 3)*(44*n - 69)*a(n) + (n - 1)*(176*n^3 - 9591*n^2 + 38703*n - 40640)*a(n-1) + (-17479*n^4 + 218005*n^3 - 959616*n^2 + 1797890*n - 1221920)*a(n-2) + 6*(3*n - 10)*(2*n - 7)*(3*n - 11)*(517*n - 1198)*a(n-3) = 0 for n >= 4. - R. J. Mathar, Nov 26 2012

Extensions

More terms from Paul Barry, Nov 05 2006

A137570 Square array, read by antidiagonals, where row n+1 equals the partial sums of the previous row after removing the terms in positions {n, n+1} from row n for n>=0, with row 0 equal to all 1's.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 6, 10, 4, 1, 1, 7, 29, 16, 5, 1, 1, 8, 36, 60, 23, 6, 1, 1, 9, 44, 186, 100, 31, 7, 1, 1, 10, 53, 230, 397, 150, 40, 8, 1, 1, 11, 63, 283, 1281, 681, 211, 50, 9, 1, 1, 12, 74, 346, 1564, 2802, 1051, 284, 61, 10, 1, 1, 13, 86, 420, 1910, 9294, 4908
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2008

Keywords

Examples

			Square array begins:
  (1),(1), 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
  1,(2),(3), 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...;
  1, 5,(10),(16), 23, 31, 40, 50, 61, 73, 86, 100, 115, 131, 148, ...;
  1, 6, 29,(60),(100), 150, 211, 284, 370, 470, 585, 716, 864, ...;
  1, 7, 36, 186,(397),(681), 1051, 1521, 2106, 2822, 3686, 4716, ...;
  1, 8, 44, 230, 1281,(2802),(4908), 7730, 11416, 16132, 22063, ...;
  1, 9, 53, 283, 1564, 9294,(20710),(36842), 58905, 88319, 126730, ...;
  1, 10, 63, 346, 1910, 11204, 70109,(158428),(285158), 461190, ...;
  1, 11, 74, 420, 2330, 13534, 83643, 544833,(1244413),(2260257), ...;
  ...
For each row, remove the terms along the diagonals (in parenthesis),
and then take partial sums to obtain the next row.
GENERATING FUNCTIONS.
The g.f. of n-th lower diagonal equals D(x)*F(x)^2*C(x)^n and
the g.f. of n-th upper diagonal equals D(x)*F(x)^n,
where D(x) is g.f. of main diagonal (A137571):
[1, 2, 10, 60, 397, 2802, 20710, 158428, 1244413, 9980220, ...]
defined by:
D(x) = 1/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where
C(x) = 1 + x*C(x)^2 is g.f. of Catalan numbers (A000108):
[1, 1, 2, 5, 14, 42, 132, 429, 1430, ..., C(2*n,n)/(n+1), ...] and
F(x) = 1 + x*F(x)^4 is g.f. of A002293:
[1, 1, 4, 22, 140, 969, 7084, 53820, ..., C(4*n,n)/(3*n+1), ...].

Crossrefs

Cf. A130523 (variant); diagonals: A137571, A137572, A137573; related: A000108, A002293.

Programs

  • PARI
    T(n, k)=if(k<0, 0, if(n==0, 1, T(n, k-1) + if(n-1>k, T(n-1, k), T(n-1, k+2))))
  • PARI
    /* Using Formula for G.F.: */ T(n,k)=local(m=max(n,k)+1,C,F,D,A); C=subst(Ser(vector(m,r,binomial(2*r-2,r-1)/r)),x,x*y); F=subst(Ser(vector(m,r,binomial(4*r-4,r-1)/(3*r-2))),x,x*y); D=1/(1-x*y*C*F^2-x*y*F^3); A=D*(1/(1-y*F) + x*C*F^2/(1-x*C)); polcoeff(polcoeff(A+O(x^m),n,x)+O(y^m),k,y)

Formula

G.f.: A(x,y) = D(x*y)*(1/(1 - y*F(x*y)) + x*C(x*y)*F(x*y)^2/(1 - x*C(x*y))), where D(x) = 1/(1 - x*C(x)*F(x)^2 - x*F(x)^3) is the g.f. of the main diagonal (A137571), C(x) = g.f. of Catalan numbers (A000108) and F(x) = g.f. of A002293; thus the g.f. of n-th lower diagonal = D(x)*F(x)^2*C(x)^n and the g.f. of n-th upper diagonal = D(x)*F(x)^n.
Showing 1-4 of 4 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.