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 A093645 (10,1) Pascal triangle. 29
 1, 10, 1, 10, 11, 1, 10, 21, 12, 1, 10, 31, 33, 13, 1, 10, 41, 64, 46, 14, 1, 10, 51, 105, 110, 60, 15, 1, 10, 61, 156, 215, 170, 75, 16, 1, 10, 71, 217, 371, 385, 245, 91, 17, 1, 10, 81, 288, 588, 756, 630, 336, 108, 18, 1, 10, 91, 369, 876, 1344, 1386, 966, 444, 126, 19, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The array F(10;n,m) gives in the columns m >= 1 the figurate numbers based on A017281, including the 12-gonal numbers A051624 (see the W. Lang link). This is the tenth member, d=10, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, A093560-5 and A093644 for d=1..9. This is an example of a Riordan triangle (see A093560 for a comment and A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group). 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) = (1+9*z)/(1-(1+x)*z). The SW-NE diagonals give A022100(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k, k), n >= 1, with n=0 value 9. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs. 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 W. Lang, First 10 rows and array of figurate numbers . FORMULA a(n, m) = F(10;n-m, m) for 0 <= m <= n, else 0, with F(10;0, 0)=1, F(10;n, 0)=10 if n >= 1 and F(10;n, m):=(10*n+m)*binomial(n+m-1, m-1)/m if m >= 1. Recursion: a(n, m)=0 if m > n, a(0, 0)=1; a(n, 0)=10 if n >= 1; a(n, m) = a(n-1, m) + a(n-1, m-1). G.f. column m (without leading zeros): (1+9*x)/(1-x)^(m+1), m >= 0. T(n, k) = C(n, k) + 9*C(n-1, k). - Philippe Deléham, Aug 28 2005 exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(10 + 21*x + 12*x^2/2! + x^3/3!) = 10 + 31*x + 64*x^2/2! + 110*x^3/3! + 170*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 EXAMPLE Triangle begins    1;   10,  1;   10, 11,  1;   10, 21, 12,  1;   ... MATHEMATICA t[0, 0] = 1; t[n_, k_] := Binomial[n, k] + 9*Binomial[n-1, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013, after Philippe Deléham *) PROG (Haskell) a093645 n k = a093645_tabl !! n !! k a093645_row n = a093645_tabl !! n a093645_tabl = [1] : iterate                (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [10, 1] -- Reinhard Zumkeller, Aug 31 2014 CROSSREFS Row sums: 1 for n=0 and A005015(n-1), n >= 1, alternating row sums are 1 for n=0, 9 for n=2 and 0 otherwise. The column sequences give for m=1..9: A017281, A051624 (12-gonal), A007587, A051799, A051880, A050406, A052254, A056125, A093646. Sequence in context: A287539 A288194 A143970 * A182620 A238880 A130858 Adjacent sequences:  A093642 A093643 A093644 * A093646 A093647 A093648 KEYWORD nonn,easy,tabl AUTHOR Wolfdieter Lang, Apr 22 2004 STATUS approved

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Last modified August 18 00:28 EDT 2018. Contains 313817 sequences. (Running on oeis4.)