A050405 -id:A050405 - cp's OEIS frontend

A052206 Partial sums of A050405.

1, 16, 100, 408, 1290, 3432, 8052, 17160, 33891, 62920, 110968, 187408, 304980, 480624, 736440, 1100784, 1609509, 2307360, 3249532, 4503400, 6150430, 8288280, 11033100, 14522040, 18915975

Offset: 0

Barry E. Williams, Jan 28 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Murray R.Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A050405.

Cf. A093644 ((9, 1) Pascal, column m=7).

LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,16,100,408,1290,3432,8052,17160},30] (* Harvey P. Dale, May 28 2018 *)

a(n) = (9n+7)*C(n+6, 6)/7.

G.f.: (1+8*x)/(1-x)^8.

A093644 (9,1) Pascal triangle.

Original entry on oeis.org

1, 9, 1, 9, 10, 1, 9, 19, 11, 1, 9, 28, 30, 12, 1, 9, 37, 58, 42, 13, 1, 9, 46, 95, 100, 55, 14, 1, 9, 55, 141, 195, 155, 69, 15, 1, 9, 64, 196, 336, 350, 224, 84, 16, 1, 9, 73, 260, 532, 686, 574, 308, 100, 17, 1, 9, 82, 333, 792, 1218, 1260, 882, 408, 117, 18, 1, 9, 91, 415

Offset: 0

Wolfdieter Lang, Apr 22 2004

The array F(9;n,m) gives in the columns m>=1 the figurate numbers based on A017173, including the 11-gonal numbers A051682 (see the W. Lang link).
This is the ninth member, d=9, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, A093560-5, for d=1..8.
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+8*z)/(1-(1+x)*z).
The SW-NE diagonals give A022099(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 8. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.
Triangle T(n,k), read by rows, given by (9,-8,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 10 2011

			Triangle begins
  [1];
  [9,  1];
  [9, 10,  1];
  [9, 19, 11,  1];
  ...

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.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Wolfdieter Lang, First 10 rows and array of figurate numbers .

Row sums: A020714(n-1), n >= 1, 1 for n=0, alternating row sums are 1 for n=0, 8 for n=2 and 0 otherwise.
The column sequences give for m=1..9: A017173, A051682 (11-gonal), A007586, A051798, A051879, A050405, A052206, A056117, A056003.
Cf. A093645 (d=10).

a093644 n k = a093644_tabl !! n !! k
a093644_row n = a093644_tabl !! n
a093644_tabl = [1] : iterate
               (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [9, 1]
-- Reinhard Zumkeller, Aug 31 2014

Join[{1},Table[Binomial[n,k]+8Binomial[n-1,k],{n,20},{k,0,n}]//Flatten] (* Harvey P. Dale, Aug 17 2024 *)

a(n, m) = F(9;n-m, m) for 0 <= m <= n, otherwise 0, with F(9;0, 0)=1, F(9;n, 0)=9 if n >= 1 and F(9;n, m):=(9*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)=9 if n >= 1; a(n, m) = a(n-1, m) + a(n-1, m-1).
G.f. column m (without leading zeros): (1+8*x)/(1-x)^(m+1), m >= 0.
T(n, k) = C(n, k) + 8*C(n-1, k). - Philippe Deléham, Aug 28 2005
Row n: Expansion of (9+x)*(1+x)^(n-1), n > 0. - Philippe Deléham, Oct 10 2011
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(9 + 19*x + 11*x^2/2! + x^3/3!) = 9 + 28*x + 58*x^2/2! + 100*x^3/3! + 155*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.: (-1-8*x)/(-1+x+x*y). - R. J. Mathar, Aug 11 2015

A051879 Partial sums of A051798.

Original entry on oeis.org

1, 14, 69, 224, 574, 1260, 2478, 4488, 7623, 12298, 19019, 28392, 41132, 58072, 80172, 108528, 144381, 189126, 244321, 311696, 393162, 490820, 606970, 744120, 904995, 1092546, 1309959, 1560664

Offset: 0

Barry E. Williams, Dec 14 1999

Convolution of triangular numbers (A000217) and 11-gonal numbers (A051682). [Bruno Berselli, Jul 21 2015]

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-16.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A051798; A000217, A051682.
Cf. A093644((9, 1) Pascal, column m=5).
Cf. A050405.

Accumulate[Table[(n+1)(n+2)(n+3)(9n+4)/24,{n,0,40}]] (* Harvey P. Dale, Aug 19 2012 *)

a(n) = C(n+4, 4)*(9n+5)/5.

G.f.: (1+8*x)/(1-x)^6.

A108684 a(n) = (n+1)(n+2)(n+3)(19n^3 + 111n^2 + 200n + 120)/720.

Original entry on oeis.org

1, 15, 93, 372, 1141, 2926, 6594, 13476, 25509, 45397, 76791, 124488, 194649, 295036, 435268, 627096, 884697, 1224987, 1667953, 2237004, 2959341, 3866346, 4993990, 6383260, 8080605, 10138401, 12615435, 15577408, 19097457, 23256696

Offset: 0

Emeric Deutsch, Jun 19 2005

Kekulé numbers for certain benzenoids.

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.233, # 10).

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

a:=n->(n+1)*(n+2)*(n+3)*(19*n^3+111*n^2+200*n+120)/720: seq(a(n),n=0..33);

Table[(n + 1) (n + 2) (n + 3) (19 n^3 + 111 n^2 + 200 n + 120)/720, {n, 0, 29}] (* or *)
CoefficientList[Series[(1 + 8 x + 9 x^2 + x^3)/(1 - x)^7, {x, 0, 29}], x] (* or *)
Table[Sum[Binomial[(n + 1 - k) + 1, 2] Apply[Subtract, Map[Binomial[# + 2, 3] &, {n + 1, k}]], {k, 0, n}], {n, 0, 29}] (* Michael De Vlieger, Jun 08 2017 *)

G.f.: (1 + 8*x + 9*x^2 + x^3)/(1-x)^7.
a(n) = Sum_{k=0...n} A000217(n+1-k) * (A000292(n+1) - A000292(k)). - J. M. Bergot, Jun 07 2017
a(n) = A050405(n) + A181888(n+1). - R. J. Mathar, Jul 22 2022

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A052206 Partial sums of A050405.

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A093644 (9,1) Pascal triangle.

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A051879 Partial sums of A051798.

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A108684 a(n) = (n+1)*(n+2)*(n+3)*(19*n^3 + 111*n^2 + 200*n + 120)/720.

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A108684 a(n) = (n+1)(n+2)(n+3)(19n^3 + 111n^2 + 200n + 120)/720.