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A125662 A convolution triangle of numbers based on A001906 (even indexed Fibonacci numbers). 8
1, 3, 1, 8, 6, 1, 21, 25, 9, 1, 55, 90, 51, 12, 1, 144, 300, 234, 86, 15, 1, 377, 954, 951, 480, 130, 18, 1, 987, 2939, 3573, 2305, 855, 183, 21, 1, 2584, 8850, 12707, 10008, 4740, 1386, 245, 24, 1, 6765, 26195, 43398, 40426, 23373, 8715, 2100, 316, 27, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Subtriangle of the triangle given by [0,3,-1/3,1/3,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,...] where DELTA is the operator defined in A084938. Unsigned version of A123965 and A124025.

Riordan array (1/(1-3*x+x^2), x/(1-3*x+x^2)). - Philippe Deléham, Feb 19 2012

A125662 = A078812*A007318 as infinite lower triangular matrices. - Philippe Deléham, Feb 19 2012

Triangle of coefficients of Chebyshev's S(n,x+3) polynomials (exponents of x in increasing order). - Philippe Deléham, Feb 19 2012

For 1<=k<=n, T(n,k) equals the number of (n-1)-length  words over {0,1,2,3} containing k-1 letters equal 3 and avoiding 01. - Milan Janjic, Dec 20 2016

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150)

Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

FORMULA

T(n,k) = T(n-1,k-1) + 3*T(n-1,k) - T(n-2,k) ; T(0,0)=1 ; T(n,k)=0 if k<0 or if k>n.

G.f.: 1/(1-3*x+x^2-y*x). - Philippe Deléham, Feb 19 2012

EXAMPLE

Triangle begins:

   1;

   3,  1;

   8,  6,  1;

  21, 25,  9,  1;

  55, 90, 51, 12,  1;

Triangle [0,3,-1/3,1/3,0,0,0,...] DELTA [1,0,0,0,0,0,...]begins:

  1;

  0,  1;

  0,  3,  1;

  0,  8,  6,  1;

  0, 21, 25,  9,  1;

  0, 55, 90, 51, 12,  1;

MATHEMATICA

With[{n = 9}, DeleteCases[#, 0] & /@ CoefficientList[Series[1/(1 - 3 x + x^2 - y x), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)

CROSSREFS

Cf. Diagonal sums: A000244(powers of 3); Row sums: A001353 (n+1); Diagonals: A001906(n+1), A001871; A000012, A008585, A062728.

Cf. Triangle of coefficients of Chebyshev's S(n,x+k) polynomials: A207824, A207823, A125662, A078812, A101950, A049310, A104562, A053122, A207815, A159764, A123967 for k = 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5 respectively.

Sequence in context: A103247 A030523 A207815 * A123965 A124025 A257488

Adjacent sequences:  A125659 A125660 A125661 * A125663 A125664 A125665

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Jan 28 2007

EXTENSIONS

a(45) corrected and a(51) added by Philippe Deléham, Feb 19 2012

STATUS

approved

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Last modified October 13 19:35 EDT 2019. Contains 327981 sequences. (Running on oeis4.)