A000579 -id:A000579 - cp's OEIS frontend

A253943 a(n) = 3*binomial(n+1,6).

3, 21, 84, 252, 630, 1386, 2772, 5148, 9009, 15015, 24024, 37128, 55692, 81396, 116280, 162792, 223839, 302841, 403788, 531300, 690690, 888030, 1130220, 1425060, 1781325, 2208843, 2718576, 3322704, 4034712, 4869480, 5843376, 6974352, 8282043, 9787869, 11515140

Offset: 5

Serhat Bulut, Jan 20 2015

nonn

For a set of integers {1,2,...,n}, a(n) is the sum of the 2 smallest elements of each subset with 5 elements, which is 3*C(n+1,6) (for n>=5), hence a(n) = 3*C(n+1,6) = 3*A000579(n+1). - Serhat Bulut, Oktay Erkan Temizkan, Jan 20 2015

			For A={1,2,3,4,5,6} subsets with 5 elements are {1,2,3,4,5}, {1,2,3,4,6}, {1,2,3,5,6}, {1,2,4,5,6}, {1,3,4,5,6}, {2,3,4,5,6}.
Sum of 2 smallest elements of each subset:
a(6) = (1+2) + (1+2) + (1+2) + (1+2) + (1+3) + (2+3) = 21 = 3*C(6+1,6) = 3*A000579(6+1).

G. C. Greubel, Table of n, a(n) for n = 5..1000
Serhat Bulut and Oktay Erkan Temizkan, Subset Sum Problem, 2015.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000579.

[3*Binomial(n+1,6): n in [5..40]]; // Vincenzo Librandi, Feb 13 2015

Drop[Plus @@ Flatten[Part[#, 1 ;; 2] & /@ Subsets[Range@ #, {5}]] & /@
  Range@ 28, 4] (* Michael De Vlieger, Jan 20 2015 *)
3 Binomial[Range[6, 29], 6] (* Michael De Vlieger, Feb 13 2015, after Alonso del Arte at A253946 *)

def A253943(n): return 3*binomial(n+1,6)
print([A253943(n) for n in range(5,51)]) # G. C. Greubel, Apr 03 2025

a(n) = 3*C(n+1,6) = 3*A000579(n+1).
From Amiram Eldar, Jan 09 2022: (Start)
Sum_{n>=5} 1/a(n) = 2/5.
Sum_{n>=5} (-1)^(n+1)/a(n) = 64*log(2) - 661/15. (End)
From G. C. Greubel, Apr 03 2025: (Start)
G.f.: 3*x^5/(1-x)^7.
E.g.f.: (3/6!)*x^5*(x+6)*exp(x). (End)

More terms from Vincenzo Librandi, Feb 13 2015

A289410 Irregular triangular array T(m,k) with m (row) >= 1 and k (column) >= 1 read by rows: number of m-digit numbers whose digit sum is k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 54, 61, 66, 69, 70, 69, 66, 61, 54, 45, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 165, 219, 279, 342, 405, 465, 519, 564, 597, 615, 615, 597, 564, 519, 465, 405, 342, 279, 219, 165, 120, 84

Offset: 1

Miquel Cerda, Jul 05 2017

The m-th row is palindromic; T(m,k) = T(m,9*m+1-k).

			The irregular triangle T(m,k) begins:
m\k  1  2  3  4  5   6   7   8   9   10   11  12   13   14  15  16  17  18  19
1    1  1  1  1  1   1   1   1   1;
2    1  2  3  4  5   6   7   8   9    9    8   7    6    5   4   3   2   1;
3    1  3  6  10 15  21  28  36  45   54   61  66   69   70  69  66  61  54 45,...;
4    1  4  10 20 35  56  84  120 165  219  279 342  405  465,...;
5    1  5  15 35 70  126 210 330 495  714  992 1330 1725,...;
6    1  6  21 56 126 252 462 792 1287 2001 2992,...;
etc.
Row m(2), column k(4) there are 4 numbers of 2-digits whose digits sum = 4: 13, 22, 31, 40.

Miquel Cerda, Rows n=1..10 of triangle, flattened

The row sums = 9*10^(m-1) = A052268(n). The row lengths = 9*m = A008591(n). The middle diagonal = A071976. (row m=3) = A071817, (row m=4) = A090579, (row m=5) = A090580, (row m=6) = A090581, (row m=7) = A278969, (row m=8) = A278971, (row m=9) = A289354, (column k=3) = A000217, (column k=4) = A000292, (column k=5) = A000332, (column k=6) = A000389, (column k=7) = A000579, (column k=8) = A000580, (column k=9) = A000581, (column k=10) = A035927.

row:= proc(m) local g; g:= normal((1 - x^10)^(m-1)*(x - x^10)/(1 - x)^m);
seq(coeff(g,x,j),j=1..9*m) end proc:
seq(row(k),k=1..5); # Robert Israel, Jul 19 2017

G.f. of row m: (1 - x^10)^(m-1)*(x - x^10)/(1 - x)^m.

G.f. as array: (1+x+x^2)*(1+x^3+x^6)*x*y/(1-y*(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9)). - Robert Israel, Jul 19 2017

Edited by Robert Israel, Jul 19 2017

A293616 Array of generalized Eulerian number triangles read by ascending antidiagonals, with m >= 0, n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 0, 1, 0, 1, 10, 0, 7, 1, 0, 1, 15, 0, 25, 4, 0, 0, 1, 21, 0, 65, 10, 0, 1, 0, 1, 28, 0, 140, 20, 0, 15, 4, 0, 1, 36, 0, 266, 35, 0, 90, 30, 1, 0, 1, 45, 0, 462, 56, 0, 350, 120, 5, 0, 0, 1, 55, 0, 750, 84, 0, 1050, 350, 15, 0, 1, 0

Offset: 0

Peter Luschny, Oct 14 2017

			Array starts:
m\j| 0   1  2     3    4  5       6       7    8  9      10      11      12
---|----------------------------------------------------------------------------
m=0| 1,  0, 0,    0,   0, 0,      0,      0,   0, 0,      0,      0,      0, ...
m=1| 1,  1, 0,    1,   1, 0,      1,      4,   1, 0,      1,     11,     11, ...
m=2| 1,  3, 0,    7,   4, 0,     15,     30,   5, 0,     31,    146,     91, ...
m=3| 1,  6, 0,   25,  10, 0,     90,    120,  15, 0,    301,    896,    406, ...
m=4| 1, 10, 0,   65,  20, 0,    350,    350,  35, 0,   1701,   3696,   1316, ...
m=5| 1, 15, 0,  140,  35, 0,   1050,    840,  70, 0,   6951,  11886,   3486, ...
m=6| 1, 21, 0,  266,  56, 0,   2646,   1764, 126, 0,  22827,  32172,   8022, ...
m=7| 1, 28, 0,  462,  84, 0,   5880,   3360, 210, 0,  63987,  76692,  16632, ...
m=8| 1, 36, 0,  750, 120, 0,  11880,   5940, 330, 0, 159027, 165792,  31812, ...
m=9| 1, 45, 0, 1155, 165, 0,  22275,   9900, 495, 0, 359502, 331617,  57057, ...
   A000217, A001296,A000292,A001297,A027789,A000332,A001298,A293610,A293611, ...
.
m\j| ...    13  14      15       16       17      18      19 20
---|----------------------------------------------------------------
m=0| ...,    0, 0,       0,       0,       0,      0,      0, 0, ...  [A000007]
m=1| ...,    1, 0,       1,      26,      66,     26,      1, 0, ...  [A173018]
m=2| ...,    6, 0,      63,     588,     868,    238,      7, 0, ...  [A062253]
m=3| ...,   21, 0,     966,    5376,    5586,   1176,     28, 0, ...  [A062254]
m=4| ...,   56, 0,    7770,   30660,   24570,   4200,     84, 0, ...  [A062255]
m=5| ...,  126, 0,   42525,  129780,   84630,  12180,    210, 0, ...
m=6| ...,  252, 0,  179487,  446292,  245322,  30492,    462, 0, ...
m=7| ...,  462, 0,  627396, 1315776,  625086,  68376,    924, 0, ...
m=8| ...,  792, 0, 1899612, 3444012, 1440582, 140712,   1716, 0, ...
m=9| ..., 1287, 0, 5135130, 8198190, 3063060, 270270,   3003, 0, ...
          A000389, A112494, A293612, A293613,A293614,A000579.
.
The parameter m runs over the triangles and j indexes the triangles by reading them by rows. Let T(m, n) denote the row [T(m, n, k) for 0 <= k <= n] and T(m) denote the triangle [T(m, n) for n >= 0]. Then for instance T(2) is the triangle A062253, T(4, 2) is row 2 of A062255 (which is [65, 20, 0]) and T(4, 2, 1) = 20.

Eric Weisstein's World of Mathematics, Nielsen Generalized Polylogarithm.

A000217(n) = T(n, 1, 0), A001296(n) = T(n, 2, 0), A000292(n) = T(n, 2, 1),
A001297(n) = T(n, 3, 0), A027789(n) = T(n, 3, 1), A000332(n) = T(n, 3, 2),
A001298(n) = T(n, 4, 0), A293610(n) = T(n, 4, 1), A293611(n) = T(n, 4, 2),
A000389(n) = T(n, 4, 3), A112494(n) = T(n, 5, 0), A293612(n) = T(n, 5, 1),
A293613(n) = T(n, 5, 2), A293614(n) = T(n, 5, 3), A000579(n) = T(n, 5, 4),
A144969(n) = T(n, 6, 0), A000580(n) = T(n, 6, 5), A000295(n) = T(1, n, 1),
A000460(n) = T(1, n, 2), A000498(n) = T(1, n, 3), A000505(n) = T(1, n, 4),
A000514(n) = T(1, n, 5), A001243(n) = T(1, n, 6), A001244(n) = T(1, n, 7),
A126646(n) = T(2, n, 0), A007820(n) = T(n, n, 0).
Cf. A173018, A062253, A062254, A062255, A293609.

A293616 := proc(m, n, k) option remember:
if m = 0 then m^n elif k < 0 or k > n then 0 elif n = 0 then 1 else
(k+m)*A293616(m,n-1,k) + (n-k)*A293616(m,n-1,k-1) + A293616(m-1,n,k) fi end:
for m in [$0..4] do for n in [$0..6] do print(seq(A293616(m, n, k), k=0..n)) od od;
# Sample uses:
A001298 := n -> A293616(n, 4, 0): A293614 := n -> A293616(n, 5, 3):
# Flatten:
a := proc(n) local w; w := proc(k) local t, s; t := 1; s := 1;
while t <= k do s := s + 1; t := t + s od; [s - 1, s - t + k] end:
seq(A293616(n - k, w(k)[1], w(k)[2]), k=0..n) end: seq(a(n), n = 0..11);

GenEulerianRow[0, n_] := Table[If[n==0 && j==0,1,0], {j,0,n}];
GenEulerianRow[m_, n_] := If[n==0,{1},Join[CoefficientList[x^(-m) (1 - x)^(n+m)
    PolyLog[-n-m, m, x] /. Log[1-x] -> 0, x], {0}]];
(* Sample use: *)
A173018Row[n_] := GenEulerianRow[1, n]; Table[A173018Row[n], {n, 0, 6}]

T(m, n, k) = (k + m)*T(m, n-1, k) + (n - k)*T(m, n-1, k-1) + T(m-1, n, k) with boundary conditions T(0, n, k) = 0^n; T(m, n, k) = 0 if k < 0 or k > n; and T(m, 0, k) = 0^k.

Let h(m, n) = x^(-m)*(1 - x)^(n + m)*PolyLog(-n - m, m, x) and p(m, n) the polynomial given by the expansion of h(m, n) after replacing log(1 - x) by 0. Then T(m, n, k) is the k-th coefficient of p(m, n) for 0 <= k < n.

A299807 Rectangular array read by antidiagonals: T(n,k) is the number of distinct sums of k complex n-th roots of 1, n >= 1, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 9, 10, 5, 1, 1, 6, 15, 16, 15, 6, 1, 1, 7, 19, 35, 25, 21, 7, 1, 1, 8, 28, 37, 70, 36, 28, 8, 1, 1, 9, 33, 84, 61, 126, 49, 36, 9, 1, 1, 10, 45, 96, 210, 91, 210, 64, 45, 10, 1, 1, 11, 51, 163, 225, 462, 127, 330, 81, 55, 11, 1, 1, 12, 66, 180, 477, 456, 924, 169, 495, 100, 66

Offset: 1

Max Alekseyev, Feb 24 2018

			Array starts:
  n=1:  1,  1,  1,   1,   1,    1,    1,    1,     1,     1,     1, ...
  n=2:  1,  2,  3,   4,   5,    6,    7,    8,     9,    10,    11, ...
  n=3:  1,  3,  6,  10,  15,   21,   28,   36,    45,    55,    66, ...
  n=4:  1,  4,  9,  16,  25,   36,   49,   64,    81,   100,   121, ...
  n=5:  1,  5, 15,  35,  70,  126,  210,  330,   495,   715,  1001, ...
  n=6:  1,  6, 19,  37,  61,   91,  127,  169,   217,   271,   331, ...
  n=7:  1,  7, 28,  84, 210,  462,  924, 1716,  3003,  5005,  8008, ...
  n=8:  1,  8, 33,  96, 225,  456,  833, 1408,  2241,  3400,  4961, ...
  n=9:  1,  9, 45, 163, 477, 1197, 2674, 5454, 10341, 18469, 31383, ...
  n=10: 1, 10, 51, 180, 501, 1131, 2221, 3951,  6531, 10201, 15231, ...
  ...

Max Alekseyev, Table of n, a(n) for n = 1..351

Rows: A000012 (n=1), A000027 (n=2), A000217 (n=3), A000290 (n=4), A000332 (n=5), A354343 (n=6), A000579 (n=7), A014820 (n=8).
Columns: A000012 (k=0), A000027 (k=1), A031940 (k=2).
Diagonal: A299754 (n=k).
Cf. A103314, A107754, A107861, A108380, A107848, A107753, A108381, A143008.

From Chai Wah Wu, May 28 2018: (Start)
The following are all conjectures.
For m >= 0, the 2^(m+1)-th row are the figurate numbers based on the 2^m-dimensional regular convex polytope with g.f.: (1+x)^(2^m-1)/(1-x)^(2^m+1).
For p prime, the n=p row corresponds to binomial(k+p-1,p-1) for k = 0,1,2,3,..., which is the maximum possible (i.e., the number of combinations with repetitions of k choices from p categories) with g.f.: 1/(1-x)^p.
(End)

Row 6 corrected by Max Alekseyev, Aug 14 2022

A333868 The number of ways to write n as the difference of two k-simplex numbers for k >= 2.

Original entry on oeis.org

1, 3, 3, 4, 5, 4, 3, 6, 7, 4, 4, 4, 5, 9, 4, 4, 5, 5, 7, 9, 4, 4, 4, 6, 4, 7, 7, 4, 7, 5, 3, 6, 6, 11, 9, 4, 4, 6, 4, 4, 6, 4, 5, 11, 5, 4, 4, 6, 6, 6, 5, 4, 7, 12, 8, 6, 4, 4, 6, 4, 4, 8, 5, 8, 9, 4, 4, 7, 8, 4, 5, 4, 5, 8, 4, 8, 9, 4, 5, 8, 4, 6, 10, 7, 4, 6

Offset: 2

Peter Kagey, Apr 08 2020

nonn

a(n) >= A001227(n) + A307666(n).
a(n) >= A003016(n) + A003016(n+1) - 2.
Records occur at indices 2, 3, 5, 6, 9, 10, 15, 35, 55, 105, 210, 1365, 2925, 3003,...

			The a(9) = 6 ways to write 9 as the difference of k-simplex numbers for k > 2 are:
C(5,  2) - C(2, 2) = 10 -  1,
C(6,  2) - C(4, 2) = 15 -  6,
C(10, 2) - C(9, 2) = 45 - 36,
C(5,  3) - C(3, 3) = 10 -  1,
C(9,  8) - C(7, 8) =  9 -  0, and
C(10, 9) - C(9, 9) = 10 -  1,
where C(n,k) = binomial(n,k) = A007318(n,k).

Peter Kagey, Table of n, a(n) for n = 2..5000

Cf. A001227, A003016, A007318, A307666.
Cf. A333822, A333836.
The k-simplex numbers for 2 <= k <= 6 are A000217 (k=2), A000292 (k=3), A000332 (k=4), A000389 (k=5), and A000579 (k=6).

A363174 Array read by rows: T(n,k) is the number of triangles inside a regular n-gon formed by intersecting line segments, considering all configurations of 3 line segments from k distinct vertices, with n >= 3, 3 <= k <= 6.

Original entry on oeis.org

1, 0, 0, 0, 4, 4, 0, 0, 10, 20, 5, 0, 20, 60, 30, 0, 35, 140, 105, 7, 56, 280, 280, 16, 84, 504, 630, 84, 120, 840, 1260, 180, 165, 1320, 2310, 462, 220, 1980, 3960, 796, 286, 2860, 6435, 1716, 364, 4004, 10010, 2856, 455, 5460, 15015, 5005, 560, 7280, 21840, 7744

Offset: 3

Paolo Xausa, May 19 2023

See Sommars and Sommars (1998) for a complete analysis of the problem.

			Array begins:
  n\k|     3     4     5     6
  ---+---------------------------
   3 |     1,    0,    0,    0;
   4 |     4,    4,    0,    0;
   5 |    10,   20,    5,    0;
   6 |    20,   60,   30,    0;
   7 |    35,  140,  105,    7;
   8 |    56,  280,  280,   16;
   9 |    84,  504,  630,   84;
  10 |   120,  840, 1260,  180;
  ...

Paolo Xausa, Table of n, a(n) for n = 3..10002 (rows 3..2502 of array, flattened).
Bjorn Poonen and Michael Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG], 1995-2006.
Steven E. Sommars and Tim Sommars, The Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon, Journal of Integer Sequences, Vol. 1 (1998), Article 98.1.5.
Paolo Xausa, Illustration of T(9,6).
Sequences formed by drawing all diagonals in regular polygon

Cf. A000579, A006561, A006600 (row sums), A260417.

Cf. A000292 (column k = 3), A033488 (column k = 4), A174002 (column k = 5), A363173 (column k = 6).

A363174list[rowmax_]:=Module[{d},d[m_,n_]:=Boole[Divisible[n,m]];Table[Binomial[n,k]If[4<=k<=5,k,1]-If[k==6&&EvenQ[n],((1/8n^2-9/8n+7/4)d[2,n]+3/4d[4,n]+(6n-106/3)d[6,n]-33d[12,n]-36d[18,n]-24d[24,n]+96d[30,n]+72d[42,n]+264d[60,n]+96d[84,n]+48d[90,n]+96d[120,n]+48d[210,n])n,0],{n,3,rowmax},{k,3,6}]];A363174list[20]

T(n,3) = binomial(n,3) = A000292(n-2).
T(n,4) = 4*binomial(n,4) = A033488(n-3).
T(n,5) = 5*binomial(n,5) = A174002(n-4), for n >= 4.
T(n,6) = binomial(n,6) = A000579(n) if n is odd, A000579(n) - A260417(n/2) if n is even.
Sum_{k=3..6} T(n,k) = A006600(n).

A081904 Second binomial transform of binomial(n+6, 6).

Original entry on oeis.org

1, 9, 60, 344, 1794, 8754, 40636, 181380, 784251, 3302451, 13598280, 54922860, 218131380, 853586100, 3296508840, 12581531064, 47510175861, 177681098205, 658665849636, 2422018974096, 8840103322374, 32044237392726, 115417729279620, 413255236888476, 1471500113899311

Offset: 0

Paul Barry, Mar 31 2003

Binomial transform of A055853 (without leading 0).

3rd binomial transform of (1,6,15,20,15,6,1,0,0,0,...).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (21,-189,945,-2835,5103,-5103, 2187).

Cf. A000579, A055853, A081905.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x)^6/(1-3*x)^7)); // G. C. Greubel, Oct 18 2018

LinearRecurrence[{21,-189,945,-2835,5103,-5103,2187}, {1,9,60,344,1794, 8754,40636}, 50] (* G. C. Greubel, Oct 18 2018 *)

x='x+O('x^30); Vec((1-2*x)^6/(1-3*x)^7) \\ G. C. Greubel, Oct 18 2018

a(n) = 3^n*(n^6 + 93*n^5 + 3055*n^4 + 44055*n^3 + 282424*n^2 + 720132*n + 524880)/524880.
G.f.: (1 - 2*x)^6/(1 - 3*x)^7.
E.g.f.: (720 + 4320*x + 5400*x^2 + 2400*x^3 + 450*x^4 + 36*x^5 + x^6)*exp(3*x) / 720. - G. C. Greubel, Oct 18 2018

A081906 Fourth binomial transform of binomial(n+6, 6).

Original entry on oeis.org

1, 11, 100, 820, 6290, 46006, 324556, 2225060, 14902075, 97873625, 632200000, 4025225000, 25307562500, 157349687500, 968628125000, 5909609375000, 35763408203125, 214838427734375, 1281885742187500, 7601284179687500, 44815856933593750, 262824523925781250, 1533738403320312500

Offset: 0

Paul Barry, Mar 31 2003

Binomial transform of A081905.

5th binomial transform of (1,6,15,20,15,6,1,0,0,0,...).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (35,-525,4375,-21875,65625,-109375,78125).

Cf. A000579, A081905.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x)^6/(1-5*x)^7)); // G. C. Greubel, Oct 17 2018

LinearRecurrence[{35, -525, 4375, -21875, 65625, -109375, 78125}, {1, 11, 100, 820, 6290, 46006, 324556}, 50] (* G. C. Greubel, Oct 17 2018 *)

x='x+O(x^30); Vec((1-4*x)^6/(1-5*x)^7) \\ G. C. Greubel, Oct 17 2018

a(n) = 5^n*(n^6 + 165*n^5 + 9535*n^4 + 238575*n^3 + 2590024*n^2 + 10661700*n + 11250000)/11250000.
G.f.: (1-4*x)^6/(1-5*x)^7.
E.g.f.: (720 + 4320*x + 5400*x^2 + 2400*x^3 + 450*x^4 + 36*x^5 + x^6)*exp(5*x) / 720. - G. C. Greubel, Oct 17 2018

A096944 Seventh column of (1,5)-Pascal triangle A096940.

Original entry on oeis.org

5, 31, 112, 308, 714, 1470, 2772, 4884, 8151, 13013, 20020, 29848, 43316, 61404, 85272, 116280, 156009, 206283, 269192, 347116, 442750, 559130, 699660, 868140, 1068795, 1306305, 1585836, 1913072, 2294248, 2736184, 3246320, 3832752

Offset: 0

Wolfdieter Lang, Jul 16 2004

If Y is a 5-subset of an n-set X then, for n>=10, a(n-10) is the number of 6-subsets of X having at most one element in common with Y. > - Milan Janjic, Dec 08 2007

Sixth column: A096943; eighth column: A096945.

G.f.: (5-4*x)/(1-x)^7.

a(n)= (n+30)*binomial(n+5, 5)/6 = 5*b(n)-4*b(n-1), with b(n):=A000579(n+6)=binomial(n+6, 6).

A103604 a(n) = binomial(n+6,6) * binomial(n+10,6).

Original entry on oeis.org

210, 3234, 25872, 144144, 630630, 2312310, 7399392, 21237216, 55747692, 135795660, 310390080, 671571264, 1385115732, 2738894004, 5216940960, 9610154400, 17178150990, 29881321470, 50707697040, 84126042000, 136704818250, 217946538810, 341398774080, 526116951360

Offset: 0

Zerinvary Lajos, Apr 22 2005

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A000579, A062264.

A103604:= func< n | Binomial(n+6,6)*Binomial(n+10,6) >;
[A103604(n): n in [0..30]]; // G. C. Greubel, Mar 05 2025

Table[Binomial[n+6,6]Binomial[n+10,6],{n,0,30}] (* or *) LinearRecurrence[ {13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{210,3234,25872, 144144,630630,2312310,7399392,21237216,55747692,135795660,310390080, 671571264,1385115732},30] (* Harvey P. Dale, Apr 18 2019 *)

a(n) = binomial(n+6,6)*binomial(n+10,6) \\ Colin Barker, Jul 01 2015

Vec(-42*(5*x^2+12*x+5)/(x-1)^13 + O(x^30)) \\ Colin Barker, Jul 01 2015

def A103604(n): return binomial(n+6,6)*binomial(n+10,6)
print([A103604(n) for n in range(31)]) # G. C. Greubel, Mar 05 2025

G.f.: 42*(5+12*x+5*x^2) / (1-x)^13. - Colin Barker, Jul 01 2015
a(n) = A000579(n+6)*A000579(n+10). - Michel Marcus, Jul 01 2015
From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 60*Pi^2 - 10445899/17640.
Sum_{n>=0} (-1)^n/a(n) = 447173/2205 - 2048*log(2)/7. (End)

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