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.

A108958 Number of unordered pairs of distinct length-n binary words having the same number of 1's.

Original entry on oeis.org

0, 1, 6, 27, 110, 430, 1652, 6307, 24054, 91866, 351692, 1350030, 5196204, 20050108, 77542376, 300507427, 1166737574, 4537436578, 17672369756, 68922740122, 269127888644, 1052047384708, 4116711169496, 16123793452942, 63205286441660, 247959232919620, 973469645715192
Offset: 1

Views

Author

Jeffrey Shallit, Jul 22 2005

Keywords

Comments

Equals row sums of triangle A143418, starting with a(2). - Gary W. Adamson, Aug 14 2008
In coupled systems of n spin 1/2 particles (magnetic resonance) where the spin state of the i-th particle can be coded as 0 (Sz_i=-1/2) or 1 (Sz_i=+1/2), number of distinct (vw). - Stanislav Sykora, Jun 07 2012
a(n) is the number of lattice paths from (0,0) to (n,n) using E(1,0) and N(0,1) as steps that horizontally cross the diagonal y = x with odd many times. For example, a(2) = 1 because there is only one path that horizontally crosses the diagonal with odd many times, namely, NEEN. - Ran Pan, Feb 01 2016

Examples

			a(3) = 6 because the pairs are {001,010}, {001,100}, {010,100}, {011,101}, {011,110}, {101,110}.

Links

Crossrefs

Cf. A005317, A011782, A088218, A143418.

Programs

  • Magma
    [Binomial(2*n,n)-(2^n+Binomial(2*n,n))/2: n in [1..30]]; // Vincenzo Librandi, Feb 01 2016
  • Maple
    with(combinat) a:= proc(n) add(binomial(binomial(n,k), 2), k=0..n) end;
  • Mathematica
    Table[Binomial[2 n, n] - (2^n + Binomial[2 n, n])/2, {n, 30}] (* Vincenzo Librandi, Feb 01 2016 *)
  • PARI
    a(n)=binomial(2*n-1,n-1)-2^(n-1) \\ Charles R Greathouse IV, Feb 01 2016
  • Python
    from math import comb
def A108958(n): return comb((n<<1)-1,n-1)-(1<Chai Wah Wu, Sep 23 2022

Formula

a(n) = Sum_{k=0..n} binomial(binomial(n, k), 2).
From Vladeta Jovovic, Jul 24 2005: (Start)
a(n) = binomial(2*n-1, n-1)-2^(n-1) = A088218(n) - A011782(n).
E.g.f.: exp(2*x)*(BesselI(0, 2*x)-1)/2. (End)
a(n) = (1/2)*Sum_{i+j>n,0<=i,j<=n} binomial(i+j,i). - Benoit Cloitre, May 26 2006
Conjecture: n*(n-2)*a(n) +2*(-3*n^2+7*n-3)*a(n-1) +4*(n-1)*(2*n-3) *a(n-2)=0. - R. J. Mathar, Apr 04 2012
a(n) = Sum_{0Mircea Merca, Apr 05 2012
a(n) = binomial(2*n,n) - A005317(n), - Ran Pan, Feb 01 2016
a(n) = 1/2*Sum_{k=1..n} binomial(n,k)^2 - binomial(n,k). - Gerry Martens, Oct 09 2022
a(n) ~ 2^(2*n-1)/sqrt(n*Pi). - Stefano Spezia, Apr 17 2024

A143420 Row sums of triangle A373101.

Original entry on oeis.org

1, 8, 55, 370, 2520, 17472, 123151, 880070, 6360706, 46402312, 341153384, 2524722928, 18789734496, 140521154048, 1055383259791, 7956220907758, 60179579570382, 456545145078408, 3472804505717170
Offset: 2

Views

Author

Gary W. Adamson, Aug 14 2008

Keywords

Comments

Each term in the sequence is a sum of tetrahedral numbers.
The underlying triangle mentioned as A143419 was lost and is now restored in A373101. - Georg Fischer, May 23 2024

Examples

			a(5) = 370 = (20 + 165 + 165 + 20) = C(6,3) + C(11,3) + C(11,8) + C(6,3).

Crossrefs

Cf. A108958 (row sums of A143418), A373101.

Programs

  • Maple
    seq(add((binomial(n,k)^3 - binomial(n,k))/6,k=1..n-1),n=2..20); # Georg Fischer, May 23 2024

Extensions

Definition changed, a(7) corrected and more terms from Georg Fischer, May 23 2024

A171379 Triangle, read by rows, T(n, k) = A059481(n,k)*(A059481(n,k) - 1)/2.

Original entry on oeis.org

0, 1, 3, 3, 15, 45, 6, 45, 190, 595, 10, 105, 595, 2415, 7875, 15, 210, 1540, 7875, 31626, 106491, 21, 378, 3486, 21945, 106491, 426426, 1471470, 28, 630, 7140, 54285, 313236, 1471470, 5887596, 20701395, 36, 990, 13530, 122265, 827541, 4507503, 20701395, 82812015, 295475895
Offset: 1

Views

Author

Roger L. Bagula, Dec 07 2009

Keywords

Comments

Row sums are: {0, 4, 63, 836, 11000, 147757, 2030217, 28435780, 404461170, 5824442504, ...}.
The sequence is the number of connections between figurate numbers A059481 as points page 25 Riordan.

Examples

			Triangle begins as:
   0;
   1,   3;
   3,  15,   45;
   6,  45,  190,   595;
  10, 105,  595,  2415,   7875;
  15, 210, 1540,  7875,  31626,  106491;
  21, 378, 3486, 21945, 106491,  426426, 1471470;
  28, 630, 7140, 54285, 313236, 1471470, 5887596, 20701395;

References

  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 25.

Links

Crossrefs

Cf. A059481, A143418.

Programs

  • GAP
    Flat(List([1..10], n-> List([1..n], k-> Binomial(Binomial(n+k-1, k), 2) ))); # G. C. Greubel, Nov 28 2019
  • Magma
    [Binomial(Binomial(n+k-1, k), 2): k in [1..n], n in [1..10]]; // G. C. Greubel, Nov 28 2019
  • Maple
    seq(seq( binomial(binomial(n+k-1, k), 2), k=1..n), n=1..10); # G. C. Greubel, Nov 28 2019
  • Mathematica
    Table[Binomial[Binomial[n+k-1, k], 2], {n,10}, {k,n}]//Flatten (* modified by G. C. Greubel, Nov 28 2019 *)
  • PARI
    T(n,k) = binomial(binomial(n+k-1, k), 2); \\ G. C. Greubel, Nov 28 2019
  • Sage
    [[binomial(binomial(n+k-1, k), 2) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Nov 28 2019

Formula

T(n,k) = binomial(n+k-1, k)*(binomial(n+k-1, k) - 1)/2.

A373101 Triangle read by rows, T(n,k) = (binomial(n,k)^3 - binomial(n,k))/6 for k=1..n-1 and n >= 2.

Original entry on oeis.org

1, 4, 4, 10, 35, 10, 20, 165, 165, 20, 35, 560, 1330, 560, 35, 56, 1540, 7140, 7140, 1540, 56, 84, 3654, 29260, 57155, 29260, 3654, 84, 120, 7770, 98770, 333375, 333375, 98770, 7770, 120, 165, 15180, 287980, 1543465, 2667126, 1543465, 287980, 15180, 165
Offset: 2

Views

Author

Georg Fischer, May 23 2024

Keywords

Comments

This triangle was mentioned in A143420 with the wrong A-number A143419.

Examples

			T(n,k) for n=2..7:
   1;
   4,    4;
  10,   35,   10;
  20,  165,  165,   20;
  35,  560, 1330,  560,   35;
  56, 1540, 7140, 7140, 1540, 56;

Crossrefs

Cf. A143418, A143420.

Programs

  • Maple
    seq(print(n,seq((binomial(n,k)^3 - binomial(n,k))/6,k=1..n-1)),n=2..10);
Showing 1-4 of 4 results.

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