A000579 -id:A000579 - cp's OEIS frontend

A323534 a(n) = Product_{k=1..n} (binomial(k-1,6) + binomial(n-k,6)).

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2551486386077798400, 4356795681519916813516800, 8378295212644383454317143654400, 17729411415388061815791372479702630400, 47314452412112353657024080317791118400000000, 160496342476959706163534573940481304027441961369600

Offset: 0

Vaclav Kotesovec, Jan 17 2019

nonn

Robert Israel, Table of n, a(n) for n = 0..115

Cf. A000579, A323425, A323496, A323497, A323533, A323535.

f:= proc(n) local k;  mul(binomial(k-1,6)+binomial(n-k,6),k=1..n) end proc:
map(f, [$0..20]); # Robert Israel, Feb 01 2019

Table[Product[Binomial[k-1,6] + Binomial[n-k,6], {k, 1, n}], {n, 0, 20}]

a(n) = prod(k=1, n, binomial(k-1, 6) + binomial(n-k, 6)); \\ Daniel Suteu, Jan 17 2019

a(n) ~ exp(-6*n + (15 - 4*sqrt(3))*Pi*(n-6)/6) * n^(6*n) / (6!)^n.

A240440 Number of ways to place 3 points on a triangular grid of side n so that they are not vertices of an equilateral triangle of any orientation.

Original entry on oeis.org

0, 0, 15, 105, 420, 1260, 3150, 6930, 13860, 25740, 45045, 75075, 120120, 185640, 278460, 406980, 581400, 813960, 1119195, 1514205, 2018940, 2656500, 3453450, 4440150, 5651100, 7125300, 8906625, 11044215, 13592880, 16613520, 20173560, 24347400, 29216880

Offset: 1

Heinrich Ludwig, Apr 08 2014

a(n) = 15 * A000579(n+3).

a(n) = A001498(n,3), the fourth column of coefficients of Bessel polynomials. - Ran Pan, Dec 03 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq. (2010) Vol. 13, Issue 4, Art. No. 10.4.4. See p=5 in the last equation on page 3.
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000579, A001498, A240441, A240442, A240439.

If one of the initial zeros is omitted, this is a row of the array in A129533.

[(n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48 : n in [1..50]]; // Wesley Ivan Hurt, Dec 03 2015

A240440:=n->(n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48; seq(A240440(n), n=1..50); # Wesley Ivan Hurt, Apr 08 2014

Table[(n+3)(n+2)(n+1)n(n-1)(n-2)/48, {n, 50}] (* Wesley Ivan Hurt, Apr 08 2014 *)
CoefficientList[Series[15 x^2/(1 - x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 19 2014 *)

Vec(15*x^3/(1-x)^7 + O(x^100)) \\ Colin Barker, Apr 18 2014

vector(100,n,(n^2-1)*(n^2-4)*(n+3)*n/48) \\ Derek Orr, Dec 24 2015

a(n) = (n+3)*(n+2)*(n+1)*n*(n-1)*(n-2)/48.
G.f.: 15*x^3 / (1-x)^7. - Colin Barker, Apr 18 2014
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7) for n>7. - Wesley Ivan Hurt, Dec 03 2015

A185509 Fourth accumulation array, T, of the natural number array A000027, by antidiagonals.

Original entry on oeis.org

1, 6, 7, 22, 41, 28, 63, 146, 161, 84, 154, 406, 561, 476, 210, 336, 966, 1526, 1631, 1176, 462, 672, 2058, 3556, 4361, 3976, 2562, 924, 1254, 4032, 7434, 9996, 10486, 8568, 5082, 1716, 2211, 7392, 14322, 20580, 23716, 22344, 16842, 9372, 3003, 3718, 12837, 25872, 39102, 48216, 49980, 43512, 30822, 16302, 5005, 6006, 21307, 44352, 69762, 90552, 100548, 96432, 79002, 53262, 27027, 8008, 9373, 34034, 72787, 118272, 159852

Offset: 1

Clark Kimberling, Jan 29 2011

See A144112 (and A185506) for the definition of rectangular sum array (aa).

Sequence is aa(aa(aa(aa(A000027)))).

			Northwest corner:
1.....6....22....63...154
7....41...146...406...966
28..161...561..1526..3556
84..476..1631..4361..9996

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000027, A185506, A185507, A185508.

Cf. A000579 (column 1), A257200 (row 1).

u[n_,k_]:=k(k+1)(k+2)(k+3)n(n+1)(n+2)(n+3)(5n^2+(6k+39)n+5k^2+9k+86)/86400
TableForm[Table[u[n,k],{n,1,10},{k,1,15}]]
Table[u[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = F*(5*n^2 + (6*k + 39)*n + 5*k^2 + 9*k + 86), where

F = k*(k+1)*(k+2)*(k+3)*n*(n+1)*(n+2)*(n+3)/86400.

A278969 Number of 7-digit numbers whose sum of digits is n.

Original entry on oeis.org

1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5004, 7995, 12306, 18312, 26418, 37038, 50568, 67353, 87648, 111573, 139068, 169863, 203463, 239148, 275988, 312873, 348558, 381723, 411048, 435303, 453438, 464653, 468448, 464653, 453438, 435303, 411048, 381723, 348558, 312873, 275988, 239148, 203463, 169863, 139068, 111573, 87648, 67353, 50568, 37038, 26418, 18312, 12306, 7995, 5004, 3003, 1716, 924, 462, 210, 84, 28, 7, 1

Offset: 1

Daniel Mondot, Dec 02 2016

There are 9000000 numbers with 7 decimal digits, the smallest being 1000000 and the largest 9999999.

Differs for n >= 10 (5004 vs 5005) from A000579(n+5) = binomial(n+5,6). - M. F. Hasler, Mar 05 2017

			a(2)=7: 1000001, 1000010, 1000100, 1001000, 1010000, 1100000, 2000000.

A071817 (3-digit numbers), A090579 (4-digit numbers), A090580 (5-digit numbers), A090581 (6-digit numbers), A278971 (8-digit numbers).

Rest@ CoefficientList[Series[(x - x^10)/(1 - x) ((1 - x^10)/(1 - x))^#, {x, 0, 9 (# + 1)}], x] &@ 6 (* or *)
Function[w, Count[w, #] & /@ Range[Max@ w]]@ Map[Total@ IntegerDigits@ # &, Range[10^#, 10^(# + 1) - 1]] &@ 6 (* Michael De Vlieger, Dec 07 2016 *)

b=vector(63, i, 0); for(n=1000000, 9999999, a=eval(Vec(Str(n))); b[sum(j=1, 7, a[j])]++); for(n=1, 63, print1(b[n], ", "))

Vec((1-x^9)*(1-x^10)^6/(1-x)^7) \\ shorter than (1-x^9)/(1-x)*((1-x^10)/(1-x))^6, but not better. - M. F. Hasler, Mar 05 2017

G.f.: (x - x^10)/(1 - x)*((1 - x^10)/(1 - x))^6. - Michael De Vlieger, Dec 07 2016

a(64-n) = a(n), 1 <= n <= 63. - M. F. Hasler, Mar 05 2017

A297178 Triangle read by rows: T(n,k) = number of partitions of genus 2 of n elements with k parts (n >= 6, 2 <= k <= n-4).

Original entry on oeis.org

1, 7, 21, 28, 210, 161, 84, 1134, 2184, 777, 210, 4410, 15330, 13713, 2835, 462, 13860, 75075, 121275, 63063, 8547, 924, 37422, 289905, 729960, 685608, 233772, 22407, 1716, 90090, 942942, 3396393, 4972968, 3063060, 738738, 52767, 3003, 198198, 2690688, 13096083, 27432405, 26342316, 11477466, 2063061, 114114

Offset: 6

N. J. A. Sloane, Dec 26 2017

			Triangle begins (see Table 3.2 in Yip's thesis):
    1;
    7,    21;
   28,   210,    161;
   84,  1134,   2184,    777;
  210,  4410,  15330,  13713,   2835;
  462, 13860,  75075, 121275,  63063,   8547;
  924, 37422, 289905, 729960, 685608, 233772, 22407;
  ...

Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 7.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus: a compendium of results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.2.6. See p. 12.
Robert Cori and G. Hetyei, Counting partitions of a fixed genus, arXiv preprint arXiv:1710.09992 [math.CO], 2017.
Martha Yip, Genus one partitions, Master Thesis, University of Waterloo, 2006.

Row sums are A297179.

First column is A000579.

T[n_,k_]:=((-6*(-2 + n)*(-1 + n) - k^2*(-13 + 5*n) + k*(-8 + n*(-9 + 5*n)))*(-4 + n)!*n!)/(1440*(-2 + k)!*k!*(-4 - k + n)!*(-k + n)!) (* Robert Coquereaux, Mar 05 2024 *)

T(n,k) = 8*gam(n-10,k-6) -4*gam(n-10,k-5) -15*gam(n-10,k-4) +10*gam(n-10,k-3) +gam(n-10,k-2) -4*gam(n-9,k-5) +39*gam(n-9,k-4) -10*gam(n-9,k-3) -4*gam(n-9,k-2) -15*gam(n-8,k-4) -10*gam(n-8,k-3) +6*gam(n-8,k-2) -4*gam(n-7,k-2) +10*gam(n-7,k-3) +gam(n-6,k-2) with gam(n,k) = (binomial(n+10,5) * binomial(n+5,k) * binomial(n+5,n-k)) / binomial(10,5) [Cori & Hetyei]. - Robert Coquereaux, Feb 12 2024

T(n,k) = ((-6*(-2 + n)*(-1 + n) - k^2*(-13 + 5*n) + k*(-8 + n*(-9 + 5*n)))*(-4 + n)!*n!) / (1440*(-2 + k)!*k!*(-4 - k + n)!*(-k + n)!). - Robert Coquereaux, Mar 05 2024

A017764 a(n) = binomial coefficient C(n,100).

Original entry on oeis.org

1, 101, 5151, 176851, 4598126, 96560646, 1705904746, 26075972546, 352025629371, 4263421511271, 46897636623981, 473239787751081, 4416904685676756, 38393094575497956, 312629484400483356, 2396826047070372396, 17376988841260199871, 119594570260437846171

Offset: 100

N. J. A. Sloane

More generally, the ordinary generating function for the binomial coefficients C(n,k) is x^k/(1 - x)^(k+1). - Ilya Gutkovskiy, Mar 21 2016

G. C. Greubel, Table of n, a(n) for n = 100..1100

Cf. similar sequences of the binomial coefficients C(n,k): A000012 (k = 0), A001477 (k = 1), A000217 (k = 2), A000292 (k = 3), A000332 (k = 4), A000389 (k = 5), A000579-A000582 (k = 6..9) A001287 (k = 10), A001288 (k = 11), A010965-A011001 (k = 12..48), A017713-A017763 (k = 49..99), this sequence (k = 100).

Cf. A001787, A242091.

[Binomial(n,100): n in [100..130]]; // G. C. Greubel, Nov 24 2017

Table[Binomial[n, 100], {n, 100, 5!}] (* Vladimir Joseph Stephan Orlovsky, Sep 25 2008 *)

a(n)=binomial(n,100) \\ Charles R Greathouse IV, Jun 28 2012

A017764_list, m = [], [1]*101
for _ in range(10**2):
    A017764_list.append(m[-1])
    for i in range(100):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

[binomial(n, 100) for n in range(100,115)] # Zerinvary Lajos, May 23 2009

G.f.: x^100/(1 - x)^101. - Ilya Gutkovskiy, Mar 21 2016
E.g.f.: x^100 * exp(x)/(100)!. - G. C. Greubel, Nov 24 2017
From Amiram Eldar, Dec 20 2020: (Start)
Sum_{n>=100} 1/a(n) = 100/99.
Sum_{n>=100} (-1)^n/a(n) = A001787(100)*log(2) - A242091(100)/99! = 63382530011411470074835160268800*log(2) - 1914409165727592211172313915606932788039791776845041612575266508424929 / 43575234518570298227833630584570189723 = 0.9902877001... (End)

A053643 a(n) = ceiling(binomial(n,6)/n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 4, 10, 21, 42, 77, 132, 215, 334, 501, 728, 1032, 1428, 1938, 2584, 3392, 4389, 5609, 7084, 8855, 10964, 13455, 16380, 19793, 23751, 28319, 33563, 39556, 46376, 54106, 62832, 72650, 83657, 95960, 109668, 124900

Offset: 1

N. J. A. Sloane, Mar 25 2000

Robert Israel, Table of n, a(n) for n = 1..10000
R. L. Graham and N. J. A. Sloane, Lower bounds for constant weight codes, IEEE Trans. Inform. Theory, 26 (1980), 37-43.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1,0,0,0,0,0,0,0,0,0,0,0,0,1,-5,10,-10,5,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,5,-10,10,-5,1,0,0,0,0,0,0,0,0,0,0,0,0,1,-5,10,-10,5,-1).

Cf. A000389, A000579, A053618.

[Ceiling(Binomial(n,6)/n) : n in [1..50]]; // Wesley Ivan Hurt, Nov 01 2015

seq(ceil(binomial(n,5)/6), n=0..50); # Zerinvary Lajos, Jan 12 2009

Table[Ceiling[Binomial[n, 6]/n], {n, 50}] (* Michael De Vlieger, Nov 01 2015 *)

vector(50, n, ceil(binomial(n, 6)/n)) \\ Altug Alkan, Nov 01 2015

[ceil(binomial(n,6)/n) for n in (1..50)] # G. C. Greubel, May 17 2019

From Robert Israel, Nov 01 2015: (Start)
a(n) = ceiling(A000389(n-1)/6).
G.f.: (x^52 -2*x^51 +4*x^50 -4*x^49 +2*x^48 +x^47 -x^46 +2*x^44 -2*x^43 +x^42 -x^41 +4*x^40 -5*x^39 +4*x^38 -2*x^37 +3*x^36 -5*x^35 +5*x^34 -2*x^33 -2*x^32 +5*x^31 -5*x^30 +2*x^29 +2*x^28 -5*x^27 +5*x^26 -2*x^25 -2*x^24 +5*x^23 -5*x^22 +2*x^21 +2*x^20 -5*x^19 +5*x^18 -3*x^17 +4*x^16 -7*x^15 +9*x^14 -7*x^13 +5*x^12 -4*x^11 +4*x^10 -2*x^9 +3*x^7 -4*x^6 +x^5 +6*x^4 -10*x^3 +9*x^2 -4*x +1)*x^6/((x -1)^6*(x +1)*(x^4 +1)*(x^2 +x +1)*(x^2 -x +1)*(x^6 +x^3 +1)*(x^6 -x^3 +1)*(x^8 -x^4 +1)*(x^24 -x^12 +1)).
(End)

A117411 Skew triangle associated to the Euler numbers.

Original entry on oeis.org

1, 0, 1, 0, -4, 1, 0, 0, -12, 1, 0, 0, 16, -24, 1, 0, 0, 0, 80, -40, 1, 0, 0, 0, -64, 240, -60, 1, 0, 0, 0, 0, -448, 560, -84, 1, 0, 0, 0, 0, 256, -1792, 1120, -112, 1, 0, 0, 0, 0, 0, 2304, -5376, 2016, -144, 1, 0, 0, 0, 0, 0, -1024, 11520, -13440, 3360, -180, 1, 0, 0, 0, 0, 0, 0, -11264, 42240, -29568, 5280, -220, 1

Offset: 0

Paul Barry, Mar 13 2006

Inverse is A117414. Row sums of the inverse are the Euler numbers A000364.

Triangle, read by rows, given by [0,-4,4,0,0,0,0,0,0,0,...] DELTA [1,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 01 2009

			Triangle begins
  1;
  0,  1;
  0, -4,   1;
  0,  0, -12,   1;
  0,  0,  16, -24,    1;
  0,  0,   0,  80,  -40,     1;
  0,  0,   0, -64,  240,   -60,      1;
  0,  0,   0,   0, -448,   560,    -84,      1;
  0,  0,   0,   0,  256, -1792,   1120,   -112,      1;
  0,  0,   0,   0,    0,  2304,  -5376,   2016,   -144,      1;
  0,  0,   0,   0,    0, -1024,  11520, -13440,   3360,   -180,    1;
  0,  0,   0,   0,    0,     0, -11264,  42240, -29568,   5280, -220,    1;
  0,  0,   0,   0,    0,     0,   4096, -67584, 126720, -59136, 7920, -264, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000364, A006495 (row sums), A098158, A117413, A117414.

Cf. A000217, A000332, A000579, A000581, A262710.

A117411:= func< n,k | (-4)^(n-k)*(&+[Binomial(n,k-j)*Binomial(j,n-k): j in [0..n-k]]) >;
[A117411(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Sep 07 2022

T[n_,k_]:= T[n,k]= (-4)^(n-k)*Sum[Binomial[n, k-j]*Binomial[j, n-k], {j,0,n-k}];
Table[T[n,k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 07 2022 *)

def A117411(n,k): return (-4)^(n-k)*sum(binomial(n,k-j)*binomial(j,n-k) for j in (0..n-k))
flatten([[A117411(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Sep 07 2022

Sum_{k=0..n} T(n, k) = A006495(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A117413(n).
T(n, k) = (-4)^(n-k)*Sum_{j=0..n-k} C(n,k-j)*C(j,n-k).
G.f.: (1-x*y)/(1-2x*y+x^2*y(y+4)). - Paul Barry, Mar 14 2006
T(n, k) = (-4)^(n-k)*A098158(n,k). - Philippe Deléham, Nov 01 2009
T(n, k) = 2*T(n-1,k-1) - 4*T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,1) = 1, T(1,0) = 0, T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Oct 31 2013
From G. C. Greubel, Sep 07 2022: (Start)
T(n, n) = 1.
T(n, n-1) = -4*A000217(n-1), n >= 1.
T(n, n-2) = (-4)^2 * A000332(n), n >= 2.
T(n, n-3) = (-4)^3 * A000579(n), n >= 3.
T(n, n-4) = (-4)^4 * A000581(n), n >= 4.
T(2*n, n) = A262710(n). (End)

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

Original entry on oeis.org

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

A175113 a(n) = ((2*n + 1)^6 + 1)/2.

Original entry on oeis.org

1, 365, 7813, 58825, 265721, 885781, 2413405, 5695313, 12068785, 23522941, 42883061, 74017945, 122070313, 193710245, 297411661, 443751841, 645733985, 919132813, 1282863205, 1759371881, 2375052121, 3160681525, 4151882813

Offset: 0

R. J. Mathar, Feb 13 2010

Convolution of the finite sequence 1, 358, 5279, 11764, 5279, 358, 1 with A000579. Partial sums of A175114.
Subsequence of A001844 because a(n)=(A050492(n+1)-1)^2+A050492(n+1)^2. - Bruno Berselli, Dec 28 2010
a(n) is also the first integer in a sum of (2*n + 1)^6 consecutive integers that equals (2*n + 1)^12. - Patrick J. McNab, Dec 26 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. J. Mathar, Point counts of D_k and some A_k and E_k integer lattices inside hypercubes, arXiv:1002.3844, Variable V_6^(g)(n).
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

I:=[1, 365, 7813, 58825, 265721, 885781, 2413405]; [n le 7 select I[n] else 7*Self(n-1) - 21*Self(n-2) + 35*Self(n-3) - 35*Self(n-4) + 21*Self(n-5) - 7*Self(n-6) + Self(n-7): n in [1..40]]; // Vincenzo Librandi, Dec 20 2012

CoefficientList[Series[(1 + 358*x + 5279*x^2 + 11764*x^3 + 5279*x^4 + 358*x^5 + x^6)/(1 - x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 20 2012 *)

a(n)= 7*a(n-1) -21*a(n-2) +35*a(n-3) -35*a(n-4) +21*a(n-5) -7*a(n-6) +a(n-7).
G.f.: (1+358*x+5279*x^2+11764*x^3+5279*x^4+358*x^5+x^6)/(1-x)^7.
a(n) = (2*n^2+2*n+1)*(16*n^4+32*n^3+20*n^2+4*n+1). - Bruno Berselli, Dec 27 2010

cp's OEIS Frontend

A323534 a(n) = Product_{k=1..n} (binomial(k-1,6) + binomial(n-k,6)).

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A240440 Number of ways to place 3 points on a triangular grid of side n so that they are not vertices of an equilateral triangle of any orientation.

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A185509 Fourth accumulation array, T, of the natural number array A000027, by antidiagonals.

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A278969 Number of 7-digit numbers whose sum of digits is n.

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A297178 Triangle read by rows: T(n,k) = number of partitions of genus 2 of n elements with k parts (n >= 6, 2 <= k <= n-4).

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A017764 a(n) = binomial coefficient C(n,100).

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A053643 a(n) = ceiling(binomial(n,6)/n).

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