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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.

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A319203 Triangular Riordan matrix T = R^(-1) for triangular Riordan matrix R = (1/(1 - x^2 - x^3), x/(1 - x^2 - x^3)) = A104578.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, -1, -2, 0, 1, 2, -2, -3, 0, 1, 5, 5, -3, -4, 0, 1, -2, 12, 9, -4, -5, 0, 1, -21, -7, 21, 14, -5, -6, 0, 1, -14, -56, -16, 32, 20, -6, -7, 0, 1, 72, -30, -108, -30, 45, 27, -7, -8, 0, 1, 138, 210, -45, -180, -50, 60, 35, -8, -9, 0, 1
Offset: 0

Views

Author

Wolfdieter Lang, Oct 29 2018

Keywords

Comments

This is the lower triangular Riordan matrix (f(t), t*f(t)), with f(t) = F^{[-1]}(t)/t, where F(x) = x/(1 - x^2 - x^3). The expansion of f(t) is given in A319201, the sequence of column k = 0.
This gives the inverse Matrix (with upper diagonals filled with 0's) of the Riordan matrix from A104578 for any finite dimension.
The row sums give A321204, and the alternating row sums give A321205.
The A- and Z-sequences of this inverse Riordan triangle of (F(x)/x, F(x)) are A = [1, 0, -1, -1] generated by 1/(F(x)/x), and Z = [0,-1, -1] generated from 1/F(x) - 1/x. See the link W. Lang link for A- and Z- sequences in A006232 with references.
For the Boas-Buck recurrence of Riordan triangles see the Aug 10 2017 remark in A046521, also for the reference. For this Bell-type triangle the sequence b is generated by B(t) = (log(f(t)))' = (1/(1/f(t) + t^2*f(t) + 2*t^3*f(t)^2) - 1)/t, and is given in A319204.

Examples

			The triangle T(n, k) begins:
n\k     0   1    2    3   4  5  6  7  8  9 10 ...
-------------------------------------------------
0:      1
1:      0   1
2:     -1   0    1
3:     -1  -2    0    1
4:      2  -2   -3    0   1
5;      5   5   -3   -4   0  1
6:     -2  12    9   -4  -5  0  1
7:    -21  -7   21   14  -5 -6  0  1
8:    -14 -56  -16   32  20 -6 -7  0  1
9:     72 -30 -108  -30  45 27 -7 -8  0  1
10:   138 210  -45 -180 -50 60 35 -8 -9  0  1
...
Recurrence from A- and Z-sequence: 5 =  T(5, 0) = -(-2 + (-3)); 9 = T(6, 2) = 5 - (- 4 + 0).
Recurrence of Boas-Buck type, with B = [0,-2,-3, 6, ...] = A319204: 9 = T(6, 2) = ((2+1)/(6-2))*(6*1 + (-3)*0 + (-2)*(-3) + 0*(-3)) = (3/4)*12 = 9.

Crossrefs

Cf. A104578, A319201, A319204, A321204, A321205.

Programs

  • Mathematica
    (* The function RiordanArray is defined in A256893. *)
nmax = 10;
R = RiordanArray[1/(1 - #^2 - #^3)&, #/(1 - #^2 - #^3)&, nmax+1];
M = Inverse[PadRight[#, nmax+1]& /@ R];
T[n_, k_] := M[[n+1, k+1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

Formula

Recurrence from the Z- and A-sequence: T(n, k) = 0 if n < k; T(0, 0) = 1;
T(n, 0) = -(T(n-1, 1) + T(n-1, 2)), for n >= 1; and T(n, m) = T(n-1, k-1) - T(n-1, k+1) - T(n-1, k+2), for n>=1 and k >= 1.
Boas-Buck recurrence with B(n) = A319204(n): T(n, k) = ((k+1)/(n-k))*Sum_{j=k..n-1} b(n-1-j)*T(j, k), for n >= 1, k = 0,1, ..., n-1, and input T(n,n) = 1, for n >= 0.
G.f. of row polynomials R(n,x) = Sum_{k=0..n} T(n, k)*x^k is G(x,z) = f(z)/(1-x*z*f(z)) with the expansion of f given in A319201.
G.f. of column sequences Gcol(k, x) = x^k*f(x)^{k+1}, for k >= 0.

A319202 a(n) is the A-sequence for the Riordan matrix R = (1/(1- x^2 - x^3), x/(1 - x^2 - x^3)) from A104578.

Original entry on oeis.org

1, 0, 1, 1, -1, -3, 0, 10, 10, -28, -70, 42, 348, 198, -1353, -2431, 3575, 15587, 702, -74698, -89726, 264214, 753236, -441864, -4308174, -2823020, 18594787, 36373695, -52468405, -249712725, -24858975, 1267523445, 1639209195, -4671244455, -14174703810
Offset: 0

Views

Author

Wolfdieter Lang, Oct 29 2018

Keywords

Comments

See the comment in A319201, and the recurrence formula for A104578 from the A- and Z-sequences.
The Z-sequence for R is given by a(n+1), n >= 0.

Crossrefs

Cf. A104578, A319201.

Formula

a(n) = [t^n] (1/f(t)), where f(t) = F^{[-1]}(t)/t, with the compositional inverse F^{[-1]}(t) of F(x) = 1/(1 - x^2 - x^3). The expansion of f is given in A319201.

A319204 Sequence used for the Boas-Buck type recurrence for Riordan triangle A319203.

Original entry on oeis.org

0, -2, -3, 6, 20, -5, -105, -98, 420, 1008, -990, -6501, -2574, 31603, 52052, -107250, -411944, 81328, 2343042, 2413456, -9883800, -25327722, 23371634, 168185131, 77113020, -835281800, -1452148815, 2847865635, 11561517870, -1613666430, -66318892875, -72637680690, 280330495200, 750725215020
Offset: 0

Views

Author

Wolfdieter Lang, Oct 29 2018

Keywords

Comments

See A319203 for the Boas-Buck type recurrence.

Examples

			a(5) = (1/6!)*[d^6/dx^6 (1 - x^2 - x^3)^6] for x = 0, which is -5.
a(5) = +15 - 20 = -5; from the sum of the signed row n=6 in A321203, with parity of e2 + e3 from A321201 even and odd.

Crossrefs

Cf. A319203, A321201, A321203.

Formula

O.g.f.: (log(f(x)))' = (1/(1/f(x) + x^2*f(x) + 2*x^3*f(x)^2) - 1)/x, with the expansion of f given in A319201. f(x) = F^{[-1]}(x)/x, where F(t) = t/(1 - t^2 - t^3).
a(n) = (1/(n+1)!)*[d^(n+1)/dx^(n+1) (1 - x^2 - x^3)^(n+1)] evaluated at x = 0, for n >= 0. (Cf. Joerg Arndt's conjecture for A176806, which is proved there.)
a(n-1) = Sum_{2*e + 3*e3 = n} (-1)^(e2+e3)*n!/((n - (e2+e3))!*e2!*e3!), n >= 2, with a(0) = 0. The pairs (e2, e3) are given in A321201; see also the multinomial coefficient table A321203 and add the sign factors.

A321204 Row sums of Riordan triangle A319203.

Original entry on oeis.org

1, 1, 0, -2, -2, 4, 11, -3, -46, -38, 152, 334, -301, -1855, -719, 7869, 12450, -23422, -87520, 12924, 447028, 458664, -1699630, -4308816, 3528771, 26104199, 12852166, -118868366, -207528889, 368534179, 1517678456, -109794866, -8068372174, -9080718086, 31552958156, 85735418464, -63813069724, -530936155928, -300127192606
Offset: 0

Views

Author

Wolfdieter Lang, Nov 09 2018

Keywords

Comments

The alternating row sums are given in A321205.

Crossrefs

Cf. A319201, A319203, A321205.

Formula

G.f: f(x)/(1 - x*f(x)) where f(x) = F^([-1])(x)/x with F^{[-1]}(y) the compositional inverse of F(x) = x/(1 - x^2 - x^3). The expansion of f is given in A319201.

A321205 Alternating row sums of Riordan triangle A319203.

Original entry on oeis.org

1, -1, 0, 0, 2, 0, -5, -7, 14, 42, -12, -198, -165, 715, 1573, -1573, -9282, -3094, 41548, 63308, -131784, -468996, 111758, 2496144, 2369851, -9926455, -23864570, 22785180, 150243015, 60480225, -709314480, -1161220320, 2325542190, 8874775170, -1613776920
Offset: 0

Views

Author

Wolfdieter Lang, Nov 09 2018

Keywords

Comments

The row sums are given in A321204.

Crossrefs

Cf. A319201, A319203, A321204.

Formula

G.f: f(x)/(1 + x*f(x)) where f(x) = F^([-1])(x)/x with F^{[-1]}(y) the compositional inverse of F(x) = x/(1 - x^2 - x^3). The expansion of f is given in A319201.
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