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.

A120668 Row sums of A129844.

Original entry on oeis.org

1, 16, 110, 156, 399, 1240
Offset: 1

Views

Author

N. J. A. Sloane, Dec 30 2008, based on email from Roger L. Bagula, May 22 2007

Keywords

Extensions

a(6) corrected by Andrey Zabolotskiy, Jul 23 2024

A054456 Convolution triangle of A000129(n) (Pell numbers).

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 12, 14, 6, 1, 29, 44, 27, 8, 1, 70, 131, 104, 44, 10, 1, 169, 376, 366, 200, 65, 12, 1, 408, 1052, 1212, 810, 340, 90, 14, 1, 985, 2888, 3842, 3032, 1555, 532, 119, 16, 1, 2378, 7813, 11784, 10716, 6482, 2709, 784, 152, 18, 1, 5741, 20892, 35223
Offset: 0

Views

Author

Wolfdieter Lang, Apr 27 2000 and May 08 2000

Keywords

Comments

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
The G.f. for the row polynomials p(n,x) (increasing powers of x) is Pell(z)/(1-x*z*Pell(z)) with Pell(x)=1/(1-2*x-x^2) = g.f. for A000129(n+1) (Pell numbers without 0).
Column sequences are A000129(n+1), A006645(n+1), A054457(n) for m=0..2.
Riordan array (1/(1-2x-x^2),x/(1-2x-x^2)). - Paul Barry, Mar 15 2005
As a Riordan array, this factors as (1/(1-x^2),x/(1-x^2))*(1/(1-2x),x/(1-2x)), [abs(A049310) times square of A007318, or A038207]. - Paul Barry, Jul 28 2005
Coefficients of polynomials defined by P(x, 0) = 1; P(x, 1) = 2 - x; P(x, n) = (2 - x)*P(x, n - 1) + P(x, n - 2). - Roger L. Bagula, Mar 24 2008
Subtriangle (obtained by dropping the first column) of the triangle given by (0, 2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 19 2013
T(n,k) is the number of words of length n over {0,1,2,3} having k letters 3 and avoiding runs of odd length of the letter 0. - Milan Janjic, Jan 14 2017

Examples

			Fourth row polynomial (n=3): p(3,x)= 12+14*x+6*x^2+x^3
Triangle begins:
{1},
{2, 1},
{5, 4, 1},
{12, 14, 6, 1},
{29, 44, 27, 8, 1},
{70, 131,104, 44, 10, 1},
{169, 376, 366, 200, 65, 12, 1},
{408, 1052, 1212, 810, 340, 90, 14, 1},
{985, 2888, 3842, 3032, 1555, 532, 119, 16, 1},
{2378, 7813, 11784, 10716, 6482, 2709, 784, 152, 18, 1},
{5741, 20892, 35223, 36248, 25235, 12432, 4396, 1104, 189, 20, 1},
The triangle (0, 2, 1/2, -1/2, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, ...) begins:
1
0, 1
0, 2, 1
0, 5, 4, 1
0, 12, 14, 6, 1
0, 29, 44, 27, 8, 1 - _Philippe Deléham_, Feb 19 2013

Links

Crossrefs

Cf. A000129. Row sums: A006190(n+1).
Cf. A129844.

Programs

  • Maple
    G := 1/(1-(x+2)*z-z^2): Gser := simplify(series(G, z = 0, 18)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 15 do seq(coeff(P[n], x, j), j = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Aug 30 2015
T := (n,k) -> `if`(n=0,1,2^(n-k)*binomial(n,k)*hypergeom([(k-n)/2,(k-n+1)/2],[-n], -1)): seq(seq(simplify(T(n,k)),k=0..n),n=0..10); # Peter Luschny, Apr 25 2016
# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, A000129); # Peter Luschny, Oct 19 2022
  • Mathematica
    P[x_, 0] := 1; P[x_, 1] := 2 - x; P[x_, n_] := P[x, n] = (2 - x) P[x, n - 1] + P[x, n - 2]; Table[Abs@ CoefficientList[P[x, n], x], {n, 0, 10}] // Flatten (* Roger L. Bagula, Mar 24 2008, edited by Michael De Vlieger, Apr 25 2018 *)

Formula

a(n, m) := ((n-m+1)*a(n, m-1) + (n+m)*a(n-1, m-1))/(4*m), n >= m >= 1, a(n, 0)= P(n+1)= A000129(n+1) (Pell numbers without P(0)), a(n, m) := 0 if n
G.f. for column m: Pell(x)*(x*Pell(x))^m, m >= 0, with Pell(x) G.f. for A000129(n+1).
Number triangle T(n, k) with T(n, 0)=A000129(n), T(1, 1)=1, T(n, k)=0 if k>n, T(n, k)=T(n-1, k-1)+T(n-2, k)+2T(n-1, k) otherwise; T(n, k)=if(k<=n, sum{j=0..floor((n-k)/2), C(n-j, k)C(n-k-j, j)2^(n-2j-k)}; - Paul Barry, Mar 15 2005
Bivariate g.f. G(x,z) = 1/[1 - (2 + x)z - z^2]. G.f. for column k = z^k/(1 - 2z - z^2)^{k+1} (k>=0). - Emeric Deutsch, Aug 30 2015
T(n,k) = 2^(n-k)*C(n,k)*hypergeom([(k-n)/2,(k-n+1)/2],[-n],-1)) for n>=1. - Peter Luschny, Apr 25 2016

A136600 Triangle of coefficients of characteristic polynomials of a special type of Cartan matrix: E_n for E_6,E_7,E_8,E_11 example M(6)/ E_6: {{2, -1, 0, 0, 0, 0}, {-1, 2, -1, 0, 0, 0}, {0, -1, 2, -1, 0, -1}, {0, 0, -1, 2, -1, 0}, {0, 0, 0, -1, 2, 0}, {0, 0, -1, 0, 0, 2}},.

Original entry on oeis.org

1, 2, -1, 4, -4, 1, 6, -11, 6, -1, 5, -20, 21, -8, 1, 4, -34, 56, -36, 10, -1, 3, -52, 125, -120, 55, -12, 1, 2, -73, 246, -329, 220, -78, 14, -1, 1, -96, 440, -784, 714, -364, 105, -16, 1, 0, -120, 730, -1679, 1992, -1364, 560, -136, 18, -1, -1, -144, 1140, -3304, 4949, -4356, 2379, -816, 171, -20, 1, -2, -167, 1694
Offset: 1

Views

Author

Roger L. Bagula, Mar 24 2008

Keywords

Comments

Row sums are: {1, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0}.
Solution for a polynomial recursion gives for higher polynomials:
p1 = Join[{1}, Table[CharacteristicPolynomial[MO[n], x], {n, 1, 12}]];
Table[Solve[{p1[[n]] - (a0*x - b0)*p1[[n - 1]] - c0*p1[[n - 2]] == 0, p1[[n + 1]] - (a0*x - b0)* p1[[n]] - c0*p1[[n - 1]] == 0, p1[[n + 2]] - (a0*x - b0)*p1[[n + 1]] - c0*p1[[n]] == 0}, {a0, b0, c0}], {n, 3, 10}];
Polynomial recursion:
P[x, n] = (2 - x)*P[x, n - 1] + P[x, n - 2]

Examples

			{1},
{2, -1},
{4, -4, 1},
{6, -11, 6, -1},
{5, -20, 21, -8, 1},
{4, -34, 56, -36, 10, -1},
{3, -52, 125, -120,55, -12, 1},
{2, -73, 246, -329, 220, -78, 14, -1},
{1, -96, 440, -784, 714, -364, 105, -16, 1},
{0, -120, 730, -1679, 1992, -1364, 560, -136, 18, -1},
{-1, -144, 1140, -3304, 4949, -4356, 2379,-816, 171, -20, 1},
{-2, -167, 1694, -6069, 11210, -12297, 8554, -3875, 1140, -210, 22, -1},
{-3, -188, 2415, -10528, 23540, -31448, 27026, -15488, 5984, -1540, 253, -24, 1}

References

  • R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8.page 139
  • E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transl, 1957
  • Sigurdur Helgasson, Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, volume 34. A. M. S. :ISBN 0-8218-2848-7, 1978

Crossrefs

Cf. A129844.

Programs

  • Mathematica
    a[n_] := 2; b[n_] := -1; c[n_] := -1; T[n_, m_, d_] := If[ n == m, a[n], If[n == m - 1 || n == m + 1 || n ==m - 3 || n == m + 3, If[n == m - 1 &&m < d, b[m - 1], If[n == m + 1 && n < d, b[n - 1], If[n == m - 3 || n == m + 3, If[n == m - 3 && m == d, c[m - 3], If[n == m + 3 && n == d, c[n - 3], 0]]]]]]] MO[d_] := Table[If[TrueQ[T[n, m, d] == Null], 0, T[n, m, d]], {n, 1, d}, {m, 1, d}]; a1 = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[MO[n], x], x], {n, 1, 12}]]' Flatten[a1]

Formula

h(n,m)=If[ n == m, a[n], If[n == m - 1 ||n == m + 1 || n == m - 3 || n == m + 3, If[n == m - 1 && m < d,b[m - 1], If[n == m + 1 && n < d, b[n - 1], If[n ==m - 3 || n == m + 3, If[n == m - 3 && m == d, c[m - 3], If[n == m + 3 && n == d, c[n - 3], 0]]]]]]] ; for n,m<=d
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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