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 1-3 of 3 results.

A158815 Triangle T(n,k) read by rows, matrix product of A046899(row-reversed) * A130595.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 13, 5, 1, 1, 46, 16, 6, 1, 1, 166, 58, 19, 7, 1, 1, 610, 211, 71, 22, 8, 1, 1, 2269, 781, 261, 85, 25, 9, 1, 1, 8518, 2920, 976, 316, 100, 28, 10, 1, 1, 32206, 11006, 3676, 1196, 376, 116, 31, 11, 1, 1
Offset: 0

Views

Author

Gary W. Adamson and Roger L. Bagula, Mar 27 2009

Keywords

Comments

The left factor of the matrix product is the triangle which starts
1;
2, 1;
6, 3, 1;
20, 10, 4, 1;
a row-reversed version of A046899, equivalent to the triangular view of the array A092392. The right factor is the inverse of the matrix A007318, which is A130595.
Swapping the two factors, A007318^(-1) * A046899(row-reversed) would generate A158793.
Riordan array (f(x), g(x)) where f(x) is the g.f. of A026641 and where g(x) is the g.f. of A000957. - Philippe Deléham, Dec 05 2009
T(n,k) is the number of nonnegative paths consisting of upsteps U=(1,1) and downsteps D=(1,-1) of length 2n with k low peaks. (A low peak has its peak vertex at height 1.) Example: T(3,1)=5 counts UDUUUU, UDUUUD, UDUUDU, UDUUDD, UUDDUD. - David Callan, Nov 21 2011
Matrix product P^2 * Q * P^(-2), where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158793 and A171243. - Peter Bala, Jul 13 2021

Examples

			The triangle starts
       1;
       1,     1;
       4,     1,     1;
      13,     5,     1,    1;
      46,    16,     6,    1,    1;
     166,    58,    19,    7,    1,   1;
     610,   211,    71,   22,    8,   1,   1;
    2269,   781,   261,   85,   25,   9,   1,  1;
    8518,  2620,   976,  316,  100,  28,  10,  1,  1;
   32206, 11006,  3676, 1196,  376, 116,  31, 11,  1, 1;
  122464, 41746, 13938, 4544, 1442, 441, 133, 34, 12, 1, 1;
  ...

Links

Crossrefs

Cf. A000984, A026641, A046899, A158793, A171243.

Programs

  • Maple
    A158815 := proc (n, k)
  add((-1)^(j+k)*binomial(2*n-j, n)*binomial(j, k), j = 0..n);
end proc:
seq(seq(A158815(n, k), k = 0..n), n = 0..10); # Peter Bala, Jul 13 2021
  • Mathematica
    T[n_,k_]:= T[n,k]= Sum[(-1)^(j+k)*Binomial[j,k]*Binomial[2*n-j,n], {j,0,n}];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 22 2021 *)
  • Sage
    def A158815(n,k): return sum( (-1)^(j+k)*binomial(2*n-j, n)*binomial(j, k) for j in (0..n) )
flatten([[A158815(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Dec 22 2021

Formula

Sum_{k=0..n} T(n,k) = A046899(n).
T(n,0) = A026641(n).
Sum_{k=0..n} T(n,k)*x^k = A026641(n), A000984(n), A001700(n), A000302(n) for x = 0, 1, 2, 3 respectively. - Philippe Deléham, Dec 03 2009
T(n, k) = Sum_{j=0..n} binomial(j, k)*binomial(2*n-j, n). - Peter Bala, Jul 13 2021

A110438 Triangular array giving the number of NSEW unit step lattice paths of length n with terminal height k subject to the following restrictions. The paths start at the origin (0,0) and take unit steps (0,1)=N(north), (0,-1)=S(south), (1,0)=E(east) and (-1,0)=W(west) such that no paths pass below the x-axis, no paths begin with W, all W steps remain on the x-axis and there are no NS steps.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 4, 3, 1, 12, 10, 7, 4, 1, 29, 25, 18, 11, 5, 1, 71, 62, 47, 30, 16, 6, 1, 175, 155, 121, 82, 47, 22, 7, 1, 434, 389, 311, 220, 135, 70, 29, 8, 1, 1082, 979, 799, 584, 378, 212, 100, 37, 9, 1, 2709, 2471, 2051, 1541, 1039, 620, 320, 138, 46, 10, 1
Offset: 0

Views

Author

Asamoah Nkwanta (Nkwanta(AT)jewel.morgan.edu), Aug 10 2005

Keywords

Comments

The row sums are the even-indexed Fibonacci numbers.
Matrix product Q^(-1) * P * Q, where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158793. - Peter Bala, Jul 14 2021

Examples

			Triangle starts:
  1;
  1,1;
  2,2,1;
  5,4,3,1;
  12,10,7,4,1;

References

  • A. Nkwanta, A Riordan matrix approach to unifying a selected class of combinatorial arrays, Congressus Numerantium, 160 (2003), pp. 33-55.
  • A. Nkwanta, A note on Riordan matrices, Contemporary Mathematics Series, AMS, 252 (1999), pp. 99-107.
  • A. Nkwanta, Lattice paths, generating functions and the Riordan group, Ph.D. Thesis, Howard University, Washington DC, 1997.

Links

Crossrefs

Row sums are A001519(n+1).
Cf. A097724, A158793.

Programs

  • Maple
    A110438 := proc (n, k)
    add((-1)^binomial(n-i+1, 2)*binomial(floor((1/2)*n+(1/2)*i), i)*add(binomial(i, j)*binomial(j, floor((1/2)*j-(1/2)*k)), j = k..i), i = 0..n);
end proc:
seq(seq(A110438(n, k), k = 0..n), n = 0..10); # Peter Bala, Jul 14 2021
  • PARI
    \\ ColGf gives g.f. of k-th column.
ColGf(k,n)={my(g=(1 - x + x^2 - sqrt(1 - 2*x - x^2 - 2*x^3 + x^4 + O(x^(n-k+3))))/(2*x^2)); (1 - x)*g/(1 - x*g)*(x*g)^k}
T(n,k) = {polcoef(ColGf(k,n), n)} \\ Andrew Howroyd, Mar 02 2023

Formula

Recurrence is d(0, 0) = 1, d(1, 0) = 1, d(n+1, 0) = 2*d(n, 0) + Sum_{j>=1} d(n-j, j), n>=1 for leftmost column and d(n+1, k) = d(n, k-1) + d(n, k) + Sum_{j>=1} d(n-j, k+j), n>=2, k>=1 and n>j; Riordan array d(n, k): (((1-z)/(2*z))*(sqrt(1+z+z^2)/sqrt(1-3*z+z^2) - 1), ((1-z+z^2)-sqrt(1-2*z-z^2-2*z^3+z^4))/(2*z)).

Extensions

Terms a(55) and beyond from Andrew Howroyd, Mar 02 2023

A171243 Riordan array (f(x), x*g(x)), f(x) is the g.f. of A126952, g(x) is the g.f. of A117641.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 21, 6, 1, 1, 93, 25, 7, 1, 1, 421, 112, 29, 8, 1, 1, 1937, 510, 132, 33, 9, 1, 1, 9017, 2357, 606, 153, 37, 10, 1, 1, 42349, 11009, 2819, 709, 175, 41, 11, 1, 1, 200277, 51840, 13233, 3324, 819, 198, 45, 12, 1, 1
Offset: 0

Views

Author

Philippe Deléham, Dec 06 2009

Keywords

Comments

Expansion of row sums of T_(x,3), T_(x,y) defined in A039599.
Matrix product P^3 * Q * P^(-3), where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158793 and A158815. - Peter Bala, Jul 13 2021

Examples

			Triangle begins:
    1;
    1,   1;
    5,   1,  1;
   21,   6,  1, 1;
   93,  25,  7, 1, 1;
  421, 112, 29, 8, 1, 1;
  ...

Crossrefs

Cf. A061554, A158793, A158815, A171224.

Formula

Sum_{k=0..n} T(n,k)*x^k = A126952(n), A126568(n), A026375(n), A026378(n+1), A000351(n) for x = 0,1,2,3,4 respectively.
Showing 1-3 of 3 results.

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