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.

A377441 Square array T(n, k) read by rising antidiagonals. Row n has the ordinary generating function (-(n*x^3-(n+1)*x^2+x) + sqrt((n*x^3-(n+1)*x^2+x)^2 - 4*(x^3-x^2)*((n+1)*x^2-x)))/(2*(x^3-x^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 2, 6, 14, 1, 1, 2, 7, 21, 42, 1, 1, 2, 8, 30, 78, 132, 1, 1, 2, 9, 41, 136, 299, 429, 1, 1, 2, 10, 54, 222, 630, 1172, 1430, 1, 1, 2, 11, 69, 342, 1221, 2959, 4677, 4862, 1, 1, 2, 12, 86, 502, 2192, 6774, 14058, 18947, 16796, 1, 1, 2, 13, 105, 708, 3687, 14129, 37853, 67472, 77746, 58786, 1, 1, 2, 14, 126, 966, 5874, 27184
Offset: 0

Views

Author

Thomas Scheuerle, Oct 28 2024

Keywords

Comments

The Hankel sequence transform of row n satisfies the Somos-4 recurrence c(k) = (c(k-1) * c(k-3) + n*c(k-2)^2) / c(k-4). All Somos-4 sequences which are beginning with 1, 1, 1, 1, n, ... will be covered, but the Hankel transform will start with the terms 1, n, ... in each case.

Examples

			The array begins:
  [0] 1, 1, 2,  5, 14,  42,  132,   429,   1430, ... = A000108
  [1] 1, 1, 2,  6, 21,  78,  299,  1172,   4677, ... = A254316
  [2] 1, 1, 2,  7, 30, 136,  630,  2959,  14058, ...
  [3] 1, 1, 2,  8, 41, 222, 1221,  6774,  37853, ...
  [4] 1, 1, 2,  9, 54, 342, 2192, 14129,  91494, ...
  [5] 1, 1, 2, 10, 69, 502, 3687, 27184, 201045, ...

Crossrefs

Cf. A377442 (extension for -n), A105633 (row -1), A152172 (row -2).
Cf. A000108 (row 0), A254316 (row 1).
Cf. A000012 (Hankel transform of row 0), A006720 (Hankel transform of row 1).
Cf. A330025 (Hankel transform of row -1), A328380 (Hankel transform of row -2).
Cf. A371965, A377443.

Programs

  • PARI
    T(n, max_k) = Vec(-2*((n+1)*x-1)/((x-1)*(n*x-1)+((n*x^2-(n+1)*x+1)^2-4*x*(x-1)*((n+1)*x-1)+O(x^max_k))^(1/2)))

Formula

The generating function A(x) of row n satisfies: 0 = (x^3 - x^2)*A(x)^2 + (n*x^3 - (n+1)*x^2 - x)*A(x) + ((n+1)*x^2 - x).
Let d(m, n) = ( d(m-3, n)*d(m-2, n) + n)/( d(m-5, n)*d(m-4, n)*d(m-3, n)^2*d(m-2, n)^2*d(m-1, n) ) for m = even and d(m, n) = 1/( d(m-1, n)*d(m-2, n) ) for m = odd with d( < 1 , n) = 1, then the generating function of row n can be expanded as continued fractions: 1/(1 - x/(1 - d(0, n)*x/(1 - d(1, n)*x/(1 - d(2, n)*x/(...))))).
d(m, n)*d(m+1, n) is a rational solution in x of the elliptic equation y^2 = -4*x^3 + ((n+1)^2 + 8)*x^2 - 2*(n+3)*x + 1. The division polynomials for multiples of the point with x = 1, correspondent to the Hankel transform of row n in the array T(n, k).
T(n, k + 2) = Sum_{j >= 0} A377443(k, j)*n^j. This polynomial starts with A000108(k+2) + A371965(k+2)*n + ..., where A371965 is known to count peaks in the set of Catalan words of length k.

A377442 Square array read by rising antidiagonals: T(n, k) = A377441(-n, k), an extension of A377441 into the domain of negative n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 2, 4, 14, 1, 1, 2, 3, 9, 42, 1, 1, 2, 2, 6, 22, 132, 1, 1, 2, 1, 5, 12, 57, 429, 1, 1, 2, 0, 6, 6, 26, 154, 1430, 1, 1, 2, -1, 9, -2, 15, 59, 429, 4862, 1, 1, 2, -2, 14, -18, 24, 24, 138, 1223, 16796, 1, 1, 2, -3, 21, -48, 77, -23, 53, 332, 3550, 58786, 1, 1, 2, -4, 30, -98, 222, -226, 102, 107, 814, 10455, 208012, 1, 1, 2
Offset: 0

Views

Author

Thomas Scheuerle, Nov 04 2024

Keywords

Comments

The main entry for this array is A377441.

Examples

			The array begins:
  [ 0] 1, 1, 2,  5, 14,  42,  132,   429,   1430, ... = A000108
  [-1] 1, 1, 2,  4,  9,  22,   57,   154,    429, ... = A105633
  [-2] 1, 1, 2,  3,  6,  12,   26,    59,    138, ... = A152172
  [-3] 1, 1, 2,  2,  5,   6,   15,    24,     53, ...
  [-4] 1, 1, 2,  1,  6,  -2,   24,   -23,    102, ...
  [-5] 1, 1, 2,  0,  9, -18,   77,  -226,    765, ...
  [-6] 1, 1, 2, -1, 14, -48,  222,  -921,   3914, ...
  [-7] 1, 1, 2, -2, 21, -98,  531, -2756,  14373, ...
Row index written as [m] is corresponding to A377441(m, k).

Crossrefs

Cf. A377441 (The main entry for this sequence).
Cf. A105633 (row -1), A152172 (row -2).
Cf. A000108 (row 0), A254316 (row 1).
Cf. A000012 (Hankel transform of row 0), A006720 (Hankel transform of row 1).
Cf. A330025 (Hankel transform of row -1), A328380 (Hankel transform of row -2).
Cf. A371965, A377443.

A377443 Triangular array T(n,k) read by rows, satisfies A377441(n, k+2) = Sum_{m=0..k} T(k, m)*n^m.

Original entry on oeis.org

2, 5, 1, 14, 6, 1, 42, 27, 8, 1, 132, 111, 45, 10, 1, 429, 441, 222, 67, 12, 1, 1430, 1728, 1029, 382, 93, 14, 1, 4862, 6733, 4608, 2005, 599, 123, 16, 1, 16796, 26181, 20199, 10018, 3495, 881, 157, 18, 1, 58786, 101763, 87270, 48445, 19188, 5641, 1236, 195, 20, 1
Offset: 0

Views

Author

Thomas Scheuerle, Nov 04 2024

Keywords

Examples

			Triangle T(n, k) starts:
[0]     2
[1]     5,      1
[2]    14,      6,     1
[3]    42,     27,     8,     1
[4]   132,    111,    45,    10,     1
[5]   429,    441,   222,    67,    12,    1
[6]  1430,   1728,  1029,   382,    93,   14,    1
[7]  4862,   6733,  4608,  2005,   599,  123,   16,   1
[8] 16796,  26181, 20199, 10018,  3495,  881,  157,  18,  1
[9] 58786, 101763, 87270, 48445, 19188, 5641, 1236, 195, 20, 1

Crossrefs

Cf. A377441, A377442.
Cf. A254316 (row sums).
Cf. A000108, A091894, A175136, A371965.

Programs

  • PARI
    A377441(n, max_k) = Vec(-2*((n+1)*x-1)/((x-1)*(n*x-1)+((n*x^2-(n+1)*x+1)^2-4*x*(x-1)*((n+1)*x-1)+O(x^max_k))^(1/2)))
T(n, k) = Vec(A377441(y, n+5)[n+3])[n-k+1]
  • PARI
    A091894(n, k) = 2^(n-2*k-1)*binomial(n-1, 2*k)*(binomial(2*k, k)/(k + 1))
A175136(n, k) = sum(m=0,(n - k)/2,A091894(n-k, m)*binomial(n-m-1, n-k))
T(n, k) = sum(m=1, n+1-k, A175136(n+2-k, n-m+2-k)*binomial(m+k-1, m-1))+(k==0)

Formula

G.f.: (-(y*x^3-(y+1)*x^2+2*x+1) + sqrt((y*x^3-(y+1)*x^2+x)^2 - 4*(x^3-x^2)*((y+1)*x^2-x)))/(2*(x^3-x^2))/x^2.
T(n, 0) = A000108(n+2).
T(n, 1) = A371965(n+2).
T(n, 2) G.f.: x^2*1/( (x - 1)^2*(1 - 4*x)^(3/2) ).
T(n, 3) G.f.: x^3*(3*x - 1)/( (x - 1)^3*(1 - 4*x)^(5/2) ).
T(n, 4) G.f.: x^4*(x^3 + (3*x - 1)^2)/( (x - 1)^4*(1 - 4*x)^(7/2) ).
T(n, 5) G.f.: x^5*(3*x^3*(3*y - 1) + (3*x - 1)^3)/( (x - 1)^5*(1 - 4*x)^(9/2) ).
T(n, 6) G.f.: x^6*(2*x^6 + 6*x^3*(3*x - 1)^2 + (3*x - 1)^4)/( (x - 1)^6*(1 - 4*x)^(11/2) ).
T(n, 7) G.f.: x^7*(10*x^6*(3*x - 1) + 10*x^3*(3*x - 1)^3 + (3*x - 1)^5)/( (x - 1)^7*(1 - 4*x)^(13/2) ).
0 = Sum_{n=0..k} T(n+k, n)*(-1)^n*binomial(k, n).
The diagonal k terms below main diagonal has G.f.: 1 + Sum_{m=1..k+1} A175136(k+2, k-m+2)*(1 - x)^k.
T(n+k, n) = Sum_{m=1..k+1} A175136(k+2, k-m+2)*binomial(m+n-1, m-1), for k > 0.
Showing 1-3 of 3 results.

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