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 A182042 Triangle T(n,k), read by rows, given by (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, 0, -3/2, 3/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. 1
 1, 0, 3, 0, 6, 9, 0, 9, 27, 27, 0, 12, 54, 108, 81, 0, 15, 90, 270, 405, 243, 0, 18, 135, 540, 1215, 1458, 729, 0, 21, 189, 945, 2835, 5103, 5103, 2187, 0, 24, 252, 1512, 5670, 13608, 20412, 17496, 6561, 0, 27, 324, 2268, 10206, 30618, 61236, 78732, 59049, 19683 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Row sums are 4^n - 1 + 0^n. Triangle of coefficients in expansion of (1+3*x)^n - 1 + 0^n. LINKS G. C. Greubel, Rows n = 0..100 of triangle, flattened FORMULA T(n,0) = 0^n; T(n,k) = binomial(n,k)*3^k for k > 0. G.f.: (1-2*x+x^2+3*y*x^2)/(1-2*x-3*y*x+x^2+3*y*x^2). T(n,k) = 2*T(n-1,k) + 3*T(n-1,k-1) - T(n-2,k) -3*T(n-2,k-1), T(0,0) = 1, T(1,0) = T(2,0) = 0, T(1,1) = 3, T(2,1) = 6, T(2,2) = 9 and T(n,k) = 0 if k < 0 or if k > n. T(n,k) = A206735(n,k)*3^k. T(n,k) = A013610(n,k) - A073424(n,k). EXAMPLE Triangle begins: 1; 0, 3; 0, 6, 9; 0, 9, 27, 27; 0, 12, 54, 108, 81; 0, 15, 90, 270, 405, 243; 0, 18, 135, 540, 1215, 1458, 729; 0, 21, 189, 945, 2835, 5103, 5103, 2187; MAPLE T:= proc(n, k) option remember; if k=n then 3^n elif k=0 then 0 else binomial(n, k)*3^k fi; end: seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Feb 17 2020 MATHEMATICA With[{m = 9}, CoefficientList[CoefficientList[Series[(1-2*x+x^2+3*y*x^2)/(1-2*x-3*y*x+x^2+3*y*x^2), {x, 0 , m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 17 2020 *) PROG (PARI) T(n, k) = if (k==0, 1, binomial(n, k)*3^k); matrix(10, 10, n, k, T(n-1, k-1)) \\ to see the triangle \\ Michel Marcus, Feb 17 2020 (Sage) @CachedFunction def T(n, k): if (k==n): return 3^n elif (k==0): return 0 else: return binomial(n, k)*3^k [[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 17 2020 CROSSREFS Cf. A000244, A007318, A013610, A193193, A193194. Sequence in context: A309604 A244009 A321482 * A011415 A216473 A198433 Adjacent sequences: A182039 A182040 A182041 * A182043 A182044 A182045 KEYWORD easy,nonn,tabl AUTHOR Philippe Deléham, Apr 07 2012 EXTENSIONS a(48) corrected by Georg Fischer, Feb 17 2020 STATUS approved

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Last modified September 29 16:48 EDT 2023. Contains 365773 sequences. (Running on oeis4.)