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 1-5 of 5 results.

A156554 The number of integer sequences of length d = 2n+1 such that the sum of the terms is 0 and the sum of the absolute values of the terms is d-1.

Original entry on oeis.org

1, 6, 110, 2562, 66222, 1815506, 51697802, 1511679210, 45076309166, 1364497268946, 41800229045610, 1292986222651646, 40317756506959050, 1265712901796074842, 39965073938276694002, 1268208750951634765562, 40419340092267053380782, 1293151592990764737265490
Offset: 0

Views

Author

W. Edwin Clark, Feb 09 2009

Keywords

Comments

Let b(n) = S(d,n) be the coordination sequence of the lattice A_d. Then this sequence is a(n) = S(2n,n). See Conway-Sloane. The sequence is defined by Couveignes et al.

Examples

			The a(1) = 6 sequences are (1,-1,0), (-1,1,0), (1,0,-1), (-1,0,1), (0,1,-1) and (0,-1,1).

Links

  • R. H. Hardin and Colin Barker, Table of n, a(n) for n = 0..300 (terms up to n=26 from R. H. Hardin)
  • J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
  • J.-M. Couveignes, T. Ezome, and R. Lercier, Elliptic periods and primality proving, arXiv:0810.2853 [math.NT], 2008-2009.

Crossrefs

a(n) = A103881(2n, n), A103882.

Programs

  • Maple
    S:=proc(d,n) add(binomial(d,k)^2*binomial(n-k+d-1,d-1),k=0..d); end proc; a:=n->S(2*n,n);
  • Mathematica
    Table[ Binomial[-1 + 3 n, -1 + 2 n] HypergeometricPFQ[{-2 n, -2 n, -n}, {1, 1 - 3 n}, 1], {n, 0, 10}]  (* Eric W. Weisstein, Feb 10 2009 *)
  • PARI
    S(d, n) = sum(k=0, d, binomial(d,k)^2*binomial(n-k+d-1, d-1));
concat(1, vector(20, n, S(2*n,n))) \\ Colin Barker, Dec 24 2015

Formula

a(n) = S(2n,n) where S(d,n) = Sum_{k=0..d} C(d,k)^2*C(n-k+d-1,d-1) from formula (22) in Conway-Sloane.
a(n) ~ (1 + sqrt(2))^(4*n + 1/2) / (2^(5/4) * Pi * n). - Vaclav Kotesovec, Apr 10 2018
From Peter Bala, Dec 19 2020: (Start)
a(n) = Sum_{k = 0..n} C(2*n,n-k)^2 * C(2*n+k-1,k).
a(n) = Sum_{k = 1..n} C(2*n, k)*C(2*n+k, k)*C(n-1,k-1) for n >= 1.
a(n) = [x^n] P(2*n,(1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial. Cf. A103882.
a(n) = C(2*n,n)^2 * hypergeom([-n, -n, 2*n], [n+1, n+1], 1).
n^2*(2*n - 1)^2*(24*n^3 - 105*n^2 + 152*n - 73)*a(n) = (3264*n^7 - 20808*n^6 + 53900*n^5 - 73159*n^4 + 55963*n^3 - 24107*n^2 + 5436*n - 504)*a(n - 1) - (2*n - 1)*(2*n - 3)*(n - 2)^2*(24*n^3 - 33*n^2 + 14*n - 2)*a(n - 2).
Conjectural: for any prime p >= 5, a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for all positive integers n and k.
More generally, if r and s are positive integers, we conjecture that the same supercongruences hold for the sequence defined by [x^(r*n)] P(s*n,(1 + x)/(1 - x)). (End)
Even more generally, we conjecture that the same supercongruences hold for the sequence defined by [x^(r*n)] (1 + x)^(A*n) * (1 - x)^(B*n) * P(s*n,(1 + x)/(1 - x)), where A and B are integers. - Peter Bala, Mar 17 2023
a(n) = 2*Sum_{k = 0..2*n-1} (-1)^(k+1)*binomial(2*n-1, k)*binomial(n+k, k)* binomial(2*n+k-1, k) for n >= 1. - Peter Bala, Sep 25 2024

Extensions

Formula incorrectly copied from A143699 removed by R. J. Mathar, Mar 11 2010

A003733 Number of spanning trees in C_5 X P_n.

Original entry on oeis.org

5, 1805, 508805, 140503005, 38720000000, 10668237057005, 2939274449134805, 809816405722655805, 223117116976138566005, 61472262298219520000000, 16936571572967914651674005, 4666290873812984282155907805, 1285636259054921313298518442805
Offset: 1

Views

Author

Frans J. Faase

Keywords

References

  • F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Links

Crossrefs

Cf. A143699.

Programs

  • Maple
    a:= n-> (Matrix(1,9, (i,j)-> [0, 5, 1805, 508805, 140503005][1+abs(j-5)]). Matrix(9, (i,j)-> if (i=j-1) then 1 elif j=1 then -[408001, 128319, 12441, 319, 1][1/2+abs(i-9/2)] *(-1)^i else 0 fi)^n)[1, 5]: seq(a(n), n=1..20); # Alois P. Heinz, Mar 28 2009
  • Mathematica
    a[n_] := (16/41)*Sinh[n*ArcCosh[(-9 - Sqrt[5])/4]]^2*Sinh[n*ArcCosh[(-9 + Sqrt[5])/4]]^2 // Round; Array[a, 20] (* Jean-François Alcover, Jan 31 2016, after Peter Bala in A143699 *)

Formula

a(n) = 319*a(n-1) - 12441*a(n-2) + 128319*a(n-3) - 408001*a(n-4) + 408001*a(n-5) - 128319*a(n-6) + 12441*a(n-7) - 319*a(n-8) + a(n-9). [Modified by Paul Raff, Oct 30 2009]
G.f.: -5*x *(1+x) *(x^6+41*x^5-998*x^4+2722*x^3-998*x^2+41*x+1) / ( (x-1)*(x^4-279*x^3+961*x^2-279*x+1) *(x^4-39*x^3+281*x^2-39*x+1) ).
a(n) = 5 * (A143699(n))^2. - R. K. Guy, Mar 11 2010

Extensions

Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009

A003779 Number of spanning trees in P_5 x P_n.

Original entry on oeis.org

1, 209, 30305, 4140081, 557568000, 74795194705, 10021992194369, 1342421467113969, 179796299139278305, 24080189412483072000, 3225041354570508955681, 431926215138756947267505, 57847355494807961811035009, 7747424602888405489208931601
Offset: 1

Views

Author

Frans J. Faase

Keywords

Comments

Also number of domino tilings of the 9 X (2n-1) rectangle with upper left corner removed. - Alois P. Heinz, Apr 14 2011
A linear divisibility sequence of order 16; a(n) divides a(m) whenever n divides m. It is the product of two 4th-order linear divisibility sequences A143699 and A241606. - Peter Bala, Apr 26 2014

References

  • F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Links

Crossrefs

A row of A116469. Bisection of A189005.
Cf. A143699, A159764, A241606.

Programs

  • Maple
    seq(resultant(simplify(ChebyshevU(4,(x-4)*(1/2))), simplify(ChebyshevU(n-1,(1/2)*x)), x), n = 1 .. 14); # Peter Bala, Apr 27 2014
  • Mathematica
    a[n_] := 256^(n-1)*Product[Sin[(h*Pi)/10]^2 + Sin[(k*Pi)/(2*n)]^2, {h, 1, 4}, {k, 1, n-1}]; Table[a[n]//Round, {n, 1, 14}] (* Jean-François Alcover, Apr 28 2014 *)
  • PARI
    Vec(-x*(x^14-1440*x^12+26752*x^11-185889*x^10+574750*x^9-708928*x^8+708928*x^6-574750*x^5+185889*x^4-26752*x^3+1440*x^2-1)/(x^16-209*x^15+11936*x^14-274208*x^13+3112032*x^12-19456019*x^11+70651107*x^10-152325888*x^9+196664896*x^8-152325888*x^7+70651107*x^6-19456019*x^5+3112032*x^4-274208*x^3+11936*x^2-209*x+1)+O(x^99)) \\ Charles R Greathouse IV, Nov 13 2015

Formula

a(n) = 209 a(n-1)
- 11936 a(n-2)
+ 274208 a(n-3)
- 3112032 a(n-4)
+ 19456019 a(n-5)
- 70651107 a(n-6)
+ 152325888 a(n-7)
- 196664896 a(n-8)
+ 152325888 a(n-9)
- 70651107 a(n-10)
+ 19456019 a(n-11)
- 3112032 a(n-12)
+ 274208 a(n-13)
- 11936 a(n-14)
+ 209 a(n-15)
- a(n-16)
[Modified by Paul Raff, Oct 30 2009]
G.f.: -x(x^14-1440x^12+26752x^11 -185889x^10+574750x^9-708928x^8 +708928x^6-574750x^5+185889x^4 -26752x^3+1440x^2-1) / (x^16-209x^15 +11936x^14 -274208x^13+3112032x^12-19456019x^11 +70651107x^10 -152325888x^9 +196664896x^8 -152325888x^7+70651107x^6 -19456019x^5 +3112032x^4-274208x^3+11936x^2-209x+1).
From Peter Bala, Apr 26 2014: (Start)
a(n) = Resultant(U(4,(x-4)/2),U(n-1,x/2)), where U(n,x) denotes the Chebyshev polynomial of the second kind. The polynomial U(4,(x-4)/2) = 209 - 232*x + 93*x^2 - 16*x^3 + x^4 (see A159764) has zeros z_1 = (9 + sqrt(5))/2, z_2 = (9 - sqrt(5))/2, z_3 = (7 + sqrt(5))/2 and z_4 = (7 - sqrt(5))/2. Thus a(n) = U(n-1,1/2*z_1)*U(n-1,1/2*z_2)*U(n-1,1/2*z_3)*U(n-1,1/2*z_4).
a(n) = A143699(n)*A241606(n). (End)

Extensions

Recurrence from Faase's web page added by N. J. A. Sloane, Feb 03 2009

A210813 Number of spanning trees in C_10 X P_n.

Original entry on oeis.org

10, 2620860, 321437558750, 34966152200584440, 3696387867279360000000, 387686455761449000565832500, 40568852698294278820875719309510, 4242420895960521871557351517779467760, 443556393051604632125747307341249759676250
Offset: 1

Views

Author

Alois P. Heinz, Mar 26 2012

Keywords

Comments

A linear divisibility sequence: Factorizes as a product of second-order and fourth-order linear divisibility sequences. See the Formula section. - Peter Bala, May 02 2014

Links

Crossrefs

10th column of A173958.
Cf. A001109, A003733, A143699, A241606.

Programs

  • Maple
    seq(expand(10*ChebyshevU(n-1,3)*( ChebyshevU(n-1,(7 + sqrt(5))/4)*ChebyshevU(n-1,(7 - sqrt(5))/4) )^2 * ( ChebyshevU(n-1,(9 + sqrt(5))/4)*ChebyshevU(n-1,(9 - sqrt(5))/4) )^2), n = 1..10); # Peter Bala, May 02 2014

Formula

From Peter Bala, May 02 2014: (Start)
a(n) = 10*U(n-1,3)*( U(n-1,(7 + sqrt(5))/4)*U(n-1,(7 - sqrt(5))/4) )^2 * ( U(n-1,(9 + sqrt(5))/4)*U(n-1,(9 - sqrt(5))/4) )^2, where U(n,x) is a Chebyshev polynomial of the second kind,
a(n) = 10*A001109(n) * A241606(n)^2 * A143699(n)^2 = 2*A001109(n) * A241606(n)^2 * A003733(n). (End)

A241606 A linear divisibility sequence of the fourth order related to A003779.

Original entry on oeis.org

1, 11, 95, 781, 6336, 51205, 413351, 3335651, 26915305, 217172736, 1752296281, 14138673395, 114079985111, 920471087701, 7426955448000, 59925473898301, 483517428660911, 3901330906652795, 31478457514091281, 253988526230055936
Offset: 1

Views

Author

Peter Bala, Apr 26 2014

Keywords

Comments

A003779, which counts spanning trees in the graph P_5 x P_n, is a linear divisibility sequence of order 16. It factors into two fourth-order linear divisibility sequences; this sequence is one of the factors, the other is A143699.
The present sequence is the case P1 = 11, P2 = 23, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy.

Links

Crossrefs

Cf. A003779, A100047, A143699.

Programs

  • Mathematica
    a[n_] := ChebyshevU[n-1, 1/4*(7-Sqrt[5])]*ChebyshevU[n-1, 1/4*(7+Sqrt[5])]; Table[a[n]//Round, {n, 1, 20}] (* Jean-François Alcover, Apr 28 2014, after Peter Bala *)

Formula

O.g.f. x*(1 - x^2)/(1 - 11*x + 25*x^2 - 11*x^3 + x^4).
a(n) = A003779(n)/A143699(n).
a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), n >= 1, where alpha = 1/4*(11 + sqrt(29)), beta = 1/4*(11 - sqrt(29)) and where T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n)= U(n-1,1/4*(7 - sqrt(5)))*U(n-1,1/4*(7 + sqrt(5))), n >= 1, where U(n,x) denotes the Chebyshev polynomial of the second kind.
a(n) = the bottom left entry of the 2X2 matrix T(n,M), where M is the 2 X 2 matrix [0, -23/4; 1, 11/2].
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences.
a(n) = 11*a(n-1) - 25*a(n-2) + 11*a(n-3) - a(n-4). - Vaclav Kotesovec, Apr 28 2014
Showing 1-5 of 5 results.

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