A000580 -id:A000580 - cp's OEIS frontend

A128629 A triangular array generated by moving Pascal sequences to prime positions and embedding new sequences at the nonprime locations. (cf. A007318 and A000040).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 9, 10, 5, 1, 1, 6, 10, 16, 15, 6, 1, 1, 5, 18, 20, 25, 21, 7, 1, 1, 8, 15, 40, 35, 36, 28, 8, 1, 1, 9, 27, 35, 75, 56, 49, 36, 9, 1

Offset: 1

Alford Arnold, Mar 29 2007

The array can be constructed by beginning with A007318 (Pascal's triangle) placing each diagonal on a prime row. The other rows are filled in by mapping the prime factorization of the row number to the known sequences on the prime rows and multiplying term by term.

			Row six begins 1 6 18 40 75 126 ... because rows two and three are
1 2 3 4 5 6 ...
1 3 6 10 15 21 ...
The array begins
1 1 1 1 1 1 1 1 1 A000012
1 2 3 4 5 6 7 8 9 A000027
1 3 6 10 15 21 28 36 45 A000217
1 4 9 16 25 36 49 64 81 A000290
1 4 10 20 35 56 84 120 165 A000292
1 6 18 40 75 126 196 288 405 A002411
1 5 15 35 70 126 210 330 495 A000332
1 8 27 64 125 216 343 512 729 A000578
1 9 36 100 225 441 784 1296 2025 A000537
1 8 30 80 175 336 588 960 1485 A002417
1 6 21 56 126 252 462 792 1287 A000389
1 12 54 160 375 756 1372 2304 3645 A019582
1 7 28 84 210 462 924 1716 3003 A000579
1 10 45 140 350 756 1470 2640 4455 A027800
1 12 60 200 525 1176 2352 4320 7425 A004302
1 16 81 256 625 1296 2401 4096 6561 A000583
1 8 36 120 330 792 1716 3432 6435 A000580
1 18 108 400 1125 2646 5488 10368 18225 A019584
1 9 45 165 495 1287 3003 6435 12870 A000581
1 16 90 320 875 2016 4116 7680 13365 A119771
1 15 90 350 1050 2646 5880 11880 22275 A001297
1 12 63 224 630 1512 3234 6336 11583 A027810
1 10 55 220 715 2002 5005 11440 24310 A000582
1 24 162 640 1875 4536 9604 18432 32805 A019583
1 16 100 400 1225 3136 7056 14400 27225 A001249
1 14 84 336 1050 2772 6468 13728 27027 A027818
1 27 216 1000 3375 9261 21952 46656 91125 A059827
1 20 135 560 1750 4536 10290 21120 40095 A085284

Cf. A000040 A007318.

Cf. A064553 (second diagonal), A080688 (second diagonal resorted).

A128629 := proc(n,m) if n = 1 then 1; elif isprime(n) then p := numtheory[pi](n) ; binomial(p+m-1,p) ; else a := 1 ; for p in ifactors(n)[2] do a := a* procname(op(1,p),m)^ op(2,p) ; od: fi; end: # R. J. Mathar, Sep 09 2009

A-number added to each row of the examples by R. J. Mathar, Sep 09 2009

A344207 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^binomial(k+6,7).

Original entry on oeis.org

1, 8, 36, 156, 330, 1080, 1716, 4512, 7101, 14080, 19448, 43776, 50388, 91248, 128160, 209910, 245157, 431424, 480700, 800800, 949806, 1339624, 1560780, 2576376, 2684190, 3768960, 4512144, 6267472, 6724520, 10046160, 10295472, 14593272, 16081065, 20604816, 23048220

Offset: 1

Ilya Gutkovskiy, May 11 2021

nonn

Cf. A000580, A001055, A050367, A255965, A329124, A344203, A344204, A344205, A344206.

A104474 a(n) = binomial(n+3,3)*binomial(n+7,3).

Original entry on oeis.org

35, 224, 840, 2400, 5775, 12320, 24024, 43680, 75075, 123200, 194480, 297024, 440895, 638400, 904400, 1256640, 1716099, 2307360, 3059000, 4004000, 5180175, 6630624, 8404200, 10556000, 13147875, 16248960, 19936224, 24295040, 29419775

Offset: 0

Zerinvary Lajos, Apr 18 2005

			a(0): C(0+3,3)*C(0+7,3) = C(3,3)*C(7,3) = 1*35 = 35.
a(7): C(7+3,3)*C(7+7,3) = C(10,3)*(14,3) = 120*364 = 43680.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000292, A000580, A062264.

[Binomial(n+3,3)*Binomial(n+7,3): n in [0..30]]; // Vincenzo Librandi, Jul 31 2015

f[n_] := Binomial[n + 3, 3]Binomial[n + 7, 3]; Table[ f[n], {n, 0, 28}] (* Robert G. Wilson v, Apr 20 2005 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{35,224,840,2400,5775,12320,24024},40] (* Harvey P. Dale, May 25 2025 *)

vector(30, n, n--; binomial(n+3,3)*binomial(n+7,3)) \\ Michel Marcus, Jul 31 2015

def A104474(n): return 140*binomial(n+7,7)//(n+4)
print([A104474(n) for n in range(31)]) # G. C. Greubel, Mar 05 2025

a(n) = (1/36)*(n+1)*(n+2)*(n+3)*(n+5)*(n+6)*(n+7).
G.f.: (35 - 21*x + 7*x^2 - x^3)/(1-x)^7. - R. J. Mathar, Nov 30 2015
a(n) = A000292(n+1)*A000292(n+5). - R. J. Mathar, Nov 30 2015
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=0} 1/a(n) = 7/200.
Sum_{n>=0} (-1)^n/a(n) = 1/40. (End)
From G. C. Greubel, Mar 05 2025: (Start)
a(n) = 140*A000580(n+7)/(n+4).
E.g.f.: (1/36)*(1260 + 6804*x + 7686*x^2 + 3102*x^3 + 531*x^4 + 39*x^5 + x^6)*exp(x). (End)

More terms from Robert G. Wilson v, Apr 20 2005

A166812 Number of n X 7 1..2 arrays containing at least one of each value, all equal values connected, rows considered as a single number in nondecreasing order, and columns considered as a single number in nondecreasing order.

Original entry on oeis.org

6, 34, 118, 328, 790, 1714, 3430, 6433, 11438, 19446, 31822, 50386, 77518, 116278, 170542, 245155, 346102, 480698, 657798, 888028, 1184038, 1560778, 2035798, 2629573, 3365854, 4272046, 5379614, 6724518, 8347678, 10295470, 12620254, 15380935, 18643558, 22481938

Offset: 1

R. H. Hardin, Oct 21 2009

nonn

			Some solutions for n=4
...1.1.1.1.2.2.2...1.1.1.1.2.2.2...1.1.1.1.1.1.1...1.1.1.1.1.1.2
...1.1.1.2.2.2.2...1.1.2.2.2.2.2...1.1.1.1.2.2.2...1.1.1.1.1.1.2
...1.2.2.2.2.2.2...1.1.2.2.2.2.2...1.1.2.2.2.2.2...1.1.1.2.2.2.2
...1.2.2.2.2.2.2...1.2.2.2.2.2.2...1.2.2.2.2.2.2...1.2.2.2.2.2.2
------
...1.1.1.1.1.1.2...1.1.1.1.1.1.2...1.1.1.1.1.2.2...1.1.1.1.1.2.2
...1.1.1.2.2.2.2...1.1.1.1.1.2.2...1.1.1.1.1.2.2...1.1.1.2.2.2.2
...1.1.2.2.2.2.2...1.2.2.2.2.2.2...1.1.1.2.2.2.2...1.2.2.2.2.2.2
...2.2.2.2.2.2.2...2.2.2.2.2.2.2...1.1.1.2.2.2.2...2.2.2.2.2.2.2

G. C. Greubel, Table of n, a(n) for n = 1..1000

a:= n-> binomial(n+7,7)-2:
seq(a(n), n=1..50);  # Alois P. Heinz, May 31 2012

Table[Binomial[n+7,7] -2, {n, 1, 100}] (* G. C. Greubel, May 24 2016 *)

a(n) = A000580(n+7)-2. - Alois P. Heinz, May 31 2012
From G. C. Greubel, May 24 2016: (Start)
G.f.: 1/(1 - x)^8 - 2/(1-x).
E.g.f.: (1/7!)*(-5040 + 35280*x + 52920*x^2 + 29400*x^3 + 7350*x^4 + 882*x^5 + 49*x^6 + x^7)*exp(x). (End)

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.

A086614 Triangle read by rows, where T(n,k) is the coefficient of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x)^2 + xyf(x,y)^2.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 10, 12, 5, 5, 20, 42, 40, 14, 6, 35, 112, 180, 140, 42, 7, 56, 252, 600, 770, 504, 132, 8, 84, 504, 1650, 3080, 3276, 1848, 429, 9, 120, 924, 3960, 10010, 15288, 13860, 6864, 1430, 10, 165, 1584, 8580, 28028, 57330, 73920, 58344, 25740

Offset: 0

Paul D. Hanna, Jul 24 2003

			Rows:
{1},
{2, 1},
{3, 4,    2},
{4, 10,  12,    5},
{5, 20,  42,   40,   14},
{6, 35, 112,  180,  140,   42},
{7, 56, 252,  600,  770,  504,  132},
{8, 84, 504, 1650, 3080, 3276, 1848, 429}, ...

T(n,n) = A000108(n).

Cf. A086615 (antidiagonal sums), A086616 (row sums), A086617, A000292 (column 1), A277935 (column 2), A000580 (column 3 divided by 5), A000582 (column 4 divided by 14).

T := (n,k) -> `if`(k=0, n+1, binomial(2*k, k-1)*binomial(n+k+1, n-k)/k):
for n from 0 to 8 do seq(T(n,k), k=0..n) od; # Peter Luschny, Jan 26 2018

T(n,k) = binomial(2*k, k-1)*binomial(n+k+1, n-k) / k for k > 0. # Peter Luschny, Jan 26 2018

A096945 Eighth column of (1,5)-Pascal triangle A096940.

Original entry on oeis.org

5, 36, 148, 456, 1170, 2640, 5412, 10296, 18447, 31460, 51480, 81328, 124644, 186048, 271320, 387600, 543609, 749892, 1019084, 1366200, 1808950, 2368080, 3067740, 3935880, 5004675, 6310980, 7896816, 9809888, 12104136, 14840320

Offset: 0

Wolfdieter Lang, Jul 16 2004

If Y is a 5-subset of an n-set X then, for n>=11, a(n-11) is the number of 7-subsets of X having at most one element in common with Y. > - Milan Janjic, Dec 08 2007

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

Seventh column: A096944; ninth column: A096946.

CoefficientList[Series[(5-4*x)/(1-x)^8,{x,0,30}],x] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{5,36,148,456,1170,2640,5412,10296},30] (* Harvey P. Dale, Aug 16 2014 *)

G.f.: (5-4*x)/(1-x)^8.
a(n)= (n+35)*binomial(n+6, 6)/7 = 5*b(n)-4*b(n-1), with b(n):=A000580(n+7)=binomial(n+7, 7).
a(0)=5, a(1)=36, a(2)=148, a(3)=456, a(4)=1170, a(5)=2640, a(6)=5412, a(7)=10296, a(n)=8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)- 28*a(n-6)+ 8*a(n-7)-a(n-8). - Harvey P. Dale, Aug 16 2014

A104670 a(n) = binomial(n+2, 2)*binomial(n+7, n).

Original entry on oeis.org

1, 24, 216, 1200, 4950, 16632, 48048, 123552, 289575, 629200, 1283568, 2482272, 4585308, 8139600, 13953600, 23193984, 37509021, 59183784, 91333000, 138138000, 205134930, 299562120, 430775280, 610740000, 854611875, 1181415456, 1614834144, 2184124096, 2925166200

Offset: 0

Zerinvary Lajos, Apr 22 2005

			If n=0 then C(2+0,2)*C(7+0,0+0) = C(2,2)*C(7,0) = 1*1 = 1;
if n=6 then C(2+6,2)*C(7+6,0+6) = C(8,2)*C(13,6) = 28*1716 = 48048.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A000217, A000580, A062190.

A104670:= func< n | Binomial(n+2,n)*Binomial(n+7,n) >;
[A104670(n): n in [0..30]]; // G. C. Greubel, Mar 01 2025

[seq(stirling2(n+1,n)*binomial(n+6,7),n=1..25)]; # Zerinvary Lajos, Dec 06 2006

a[n_] := Binomial[n + 2, 2] * Binomial[n + 7, 7]; Array[a, 25, 0] (* Amiram Eldar, Aug 30 2022 *)

def A104670(n): return binomial(n+2,n)*binomial(n+7,n)
print([A104670(n) for n in range(31)]) # G. C. Greubel, Mar 01 2025

G.f.: (1 + 14*x + 21*x^2)/(1-x)^10. - Colin Barker, Mar 18 2012
From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=0} 1/a(n) = 49*Pi^2/3 - 288281/1800.
Sum_{n>=0} (-1)^n/a(n) = 448*log(2)/3 - 35*Pi^2/6 - 1799/40. (End)

Corrected and extended by Don Reble, Nov 21 2006

A116082 a(n) = C(n,7) + C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).

Original entry on oeis.org

0, 1, 3, 7, 15, 31, 63, 127, 254, 501, 967, 1815, 3301, 5811, 9907, 16383, 26332, 41225, 63003, 94183, 137979, 198439, 280599, 390655, 536154, 726205, 971711, 1285623, 1683217, 2182395, 2804011, 3572223, 4514872, 5663889, 7055731, 8731847

Offset: 0

Jonathan Vos Post, Mar 13 2006

Number of compositions with at most three parts distinct from 1 and with a sum at most n. - Beimar Naranjo, Mar 12 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Michael Boardman, The Egg-Drop Numbers, Mathematics Magazine, 77 (2004), 368-372. See Table 2 on page 370. [_Parthasarathy Nambi_, Jun 18 2009]
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A000217, A000225, A004006, A055795, A057703, A115567, A116082. [Parthasarathy Nambi, Jun 18 2009]

[n*(n^6-14*n^5+112*n^4-350*n^3+1099*n^2+364*n+3828)/5040: n in [0..40]]; // Vincenzo Librandi, Jun 21 2011

a:=n->n*(n^6-14*n^5+112*n^4-350*n^3+1099*n^2+364*n+3828)/5040: seq(a(n),n=0..35); # Emeric Deutsch, Apr 14 2006
seq(sum(binomial(n,k),k=1..7),n=0..35); # Zerinvary Lajos, Dec 14 2007

Table[Total[Binomial[n,Range[7]]],{n,0,40}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{0,1,3,7,15,31,63,127},41](* Harvey P. Dale, Aug 05 2011 *)

for(n=0,30, print1(n*(n^6 -14*n^5 +112*n^4 -350*n^3 +1099*n^2 +364*n +3828)/5040, ", ")) \\ G. C. Greubel, Nov 25 2017

a(n) = A000580(n) + A000579(n) + A000389(n) + A000332(n) + A000292(n) + A000217(n) + n.
a(n) = A000580(n) + A115567(n).
a(n) = n*(n^6 - 14*n^5 + 112*n^4 - 350*n^3 + 1099*n^2 + 364*n + 3828)/5040. - Emeric Deutsch, Apr 14 2006
G.f.: x*(1 - 5*x + 11*x^2 - 13*x^3 + 9*x^4 - 3*x^5 + x^6)/(1-x)^8. - R. J. Mathar, Jun 20 2011
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8), with a(0)=0, a(1)=1, a(2)=3, a(3)=7, a(4)=15, a(5)=31, a(6)=63, a(7)=127. - Harvey P. Dale, Aug 05 2011

cp's OEIS Frontend

A128629 A triangular array generated by moving Pascal sequences to prime positions and embedding new sequences at the nonprime locations. (cf. A007318 and A000040).

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A344207 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^binomial(k+6,7).

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A104474 a(n) = binomial(n+3,3)*binomial(n+7,3).

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A166812 Number of n X 7 1..2 arrays containing at least one of each value, all equal values connected, rows considered as a single number in nondecreasing order, and columns considered as a single number in nondecreasing order.

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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.

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

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A086614 Triangle read by rows, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x)^2 + xy*f(x,y)^2.

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A096945 Eighth column of (1,5)-Pascal triangle A096940.

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A086614 Triangle read by rows, where T(n,k) is the coefficient of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x)^2 + xyf(x,y)^2.