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 A093560 (3,1) Pascal triangle. 21
 1, 3, 1, 3, 4, 1, 3, 7, 5, 1, 3, 10, 12, 6, 1, 3, 13, 22, 18, 7, 1, 3, 16, 35, 40, 25, 8, 1, 3, 19, 51, 75, 65, 33, 9, 1, 3, 22, 70, 126, 140, 98, 42, 10, 1, 3, 25, 92, 196, 266, 238, 140, 52, 11, 1, 3, 28, 117, 288, 462, 504, 378, 192, 63, 12, 1, 3, 31, 145, 405, 750, 966, 882, 570, 255, 75, 13, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The array F(3;n,m) gives in the columns m >= 1 the figurate numbers based on A016777, including the pentagonal numbers A000326 (see the W. Lang link). This is the third member, d=3, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal (d=1), A029653 (d=2). This is an example of a Riordan triangle (see A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group) with o.g.f. of column nr. m of the type g(x)*(x*f(x))^m with f(0)=1. Therefore the o.g.f. for the row polynomials p(n,x):=Sum_{m=0..n} a(n,m)*x^m is G(z,x)=g(z)/(1-x*z*f(z)). Here: g(x)=(1+2*x)/(1-x), f(x)=1/(1-x), hence G(z,x)=(1+2*z)/(1-(1+x)*z). The SW-NE diagonals give the Lucas numbers A000032: L(n) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with L(0)=2. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs. Triangle T(n,k), read by rows, given by [3,-2,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 17 2009 For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013 From Wolfdieter Lang, Jan 09 2015: (Start) The signed lower triangular matrix (-1)^(n-1)*a(n,m) is the inverse of the Riordan matrix A106516; that is Riordan ((1-2*x)/(1+x),x/(1+x)). See the Peter Bala comment from Dec 23 2014 in A106516 for general Riordan triangles of the type (g(x), x/(1-x)): exp(x)*r(n,x) = d(n,x) with the e.g.f. r(n,x) of row n and the e.g.f. of diagonal n. Similarly, for general Riordan triangles of the type (g(x), x/(1+x)): exp(x)*r(n,-x) = d(n,x). (End) The n-th row polynomial is (3 + x)*(1 + x)^(n-1) for n >= 1. More generally, the n-th row polynomial of the Riordan array ( (1-a*x)/(1-b*x), x/(1-b*x) ) is (b - a + x)*(b + x)^(n-1) for n >= 1. - Peter Bala, Mar 02 2018 Binomial(n-2,k)+2*Binomial(n-3,k) is also the number of permutations avoiding both 123 and 132 with k double descents, i.e., positions with w[i]>w[i+1]>w[i+2]. - Lara Pudwell, Dec 19 2018 REFERENCES Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen. Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch.5, pp. 109-122. LINKS Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv preprint arXiv:1812.07112 [math.CO], 2018. W. Lang, First 10 rows and array of figurate numbers . FORMULA a(n, m)=F(3;n-m, m) for 0<= m <= n, otherwise 0, with F(3;0, 0)=1, F(3;n, 0)=3 if n>=1 and F(3;n, m):=(3*n+m)*binomial(n+m-1, m-1)/m if m>=1. G.f. column m (without leading zeros): (1+2*x)/(1-x)^(m+1), m>=0. Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=3 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1). T(n, k) = C(n, k) + 2*C(n-1, k). - Philippe Deléham, Aug 28 2005 Equals M * A007318, where M = an infinite triangular matrix with all 1's in the main diagonal and all 2's in the subdiagonal. - Gary W. Adamson, Dec 01 2007 Sum_{k=0..n} T(n,k) = A151821(n+1). - Philippe Deléham, Sep 17 2009 exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(3 + 7*x + 5*x^2/2! + x^3/3!) = 3 + 10*x + 22*x^2/2! + 40*x^3/3! + 65*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014 G.f.: (-1-2*x)/(-1+x+x*y). - R. J. Mathar, Aug 11 2015 EXAMPLE Triangle begins   1,   3,  1,   3,  4,  1,   3,  7,  5,   1,   3, 10, 12,   6,   1,   3, 13, 22,  18,   7,   1,   3, 16, 35,  40,  25,   8,   1,   3, 19, 51,  75,  65,  33,   9,  1,   3, 22, 70, 126, 140,  98,  42, 10,  1,   3, 25, 92, 196, 266, 238, 140, 52, 11, 1, PROG (Haskell) a093560 n k = a093560_tabl !! n !! k a093560_row n = a093560_tabl !! n a093560_tabl =  : iterate                (\row -> zipWith (+) ( ++ row) (row ++ )) [3, 1] -- Reinhard Zumkeller, Aug 31 2014 (GAP) Concatenation(, Flat(List([1..11], n->List([0..n], k->Binomial(n, k)+2*Binomial(n-1, k))))); # Muniru A Asiru, Dec 20 2018 CROSSREFS Cf. Column sequences for m=1..9: A016777, A000326 (pentagonal), A002411, A001296, A051836, A051923, A050494, A053367, A053310; A007318 (Pascal's triangle), A029653 ((2,1) Pascal triangle), A093561 ((4,1) Pascal triangle), A228196, A228576. Sequence in context: A308690 A329512 A064884 * A173934 A131504 A008311 Adjacent sequences:  A093557 A093558 A093559 * A093561 A093562 A093563 KEYWORD nonn,tabl,easy AUTHOR Wolfdieter Lang, Apr 22 2004 EXTENSIONS Incorrect connection with A046055 deleted by N. J. A. Sloane, Jul 08 2009 STATUS approved

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Last modified April 14 03:13 EDT 2021. Contains 342941 sequences. (Running on oeis4.)