A000582 -id:A000582 - cp's OEIS frontend

A097300 Tenth column (m=9) of (1,6)-Pascal triangle A096956.

6, 55, 280, 1045, 3190, 8437, 20020, 43615, 88660, 170170, 311168, 545870, 923780, 1514870, 2416040, 3759074, 5720330, 8532425, 12498200, 18007275, 25555530, 35767875, 49424700, 67492425, 91158600, 121872036, 161388480, 211822380

Offset: 0

Wolfdieter Lang, Aug 13 2004

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A000582, A096956.

Cf. other columns: A096957 (m = 3), A096958 (m = 4), A096959 (m = 5), A097297 (m = 6), A097298 (m = 7), A097299 (m = 8).

A097300[n_] := (n + 54)*Binomial[n + 8, 8]/9;
Array[A097300, 50, 0] (* Paolo Xausa, May 02 2025 *)

a(n) = A096956(n+9, 9) = 6*b(n) - 5*b(n-1) = (n+54)*binomial(n+8, 8)/9, with b(n) = A000582(n+9) = binomial(n+9, 9).

G.f.: (6-5*x)/(1-x)^10.

A101095 Fourth difference of fifth powers (A000584).

Original entry on oeis.org

1, 28, 121, 240, 360, 480, 600, 720, 840, 960, 1080, 1200, 1320, 1440, 1560, 1680, 1800, 1920, 2040, 2160, 2280, 2400, 2520, 2640, 2760, 2880, 3000, 3120, 3240, 3360, 3480, 3600, 3720, 3840, 3960, 4080, 4200, 4320, 4440, 4560, 4680, 4800, 4920, 5040, 5160, 5280

Offset: 1

Cecilia Rossiter, Dec 15 2004

Original Name: Shells (nexus numbers) of shells of shells of shells of the power of 5.

The (Worpitzky/Euler/Pascal Cube) "MagicNKZ" algorithm is: MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n + 1 - z, j)*(k - j + 1)^n, with k>=0, n>=1, z>=0. MagicNKZ is used to generate the n-th accumulation sequence of the z-th row of the Euler Triangle (A008292). For example, MagicNKZ(3,k,0) is the 3rd row of the Euler Triangle (followed by zeros) and MagicNKZ(10,k,1) is the partial sums of the 10th row of the Euler Triangle. This sequence is MagicNKZ(5,k-1,2).

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
D. J. Pengelley, The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference [pdf], Kristiansand, 2002, (ed. Otto Bekken et al), National Center for Mathematics Education, University of Gothenburg, Sweden, in press.
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Archive Machine link]
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]
Eric Weisstein, Link to section of MathWorld: Worpitzky's Identity of 1883
Eric Weisstein, Link to section of MathWorld: Eulerian Number
Eric Weisstein, Link to section of MathWorld: Nexus number
Eric Weisstein, Link to section of MathWorld: Finite Differences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Fourth differences of A000584, third differences of A022521, second differences of A101098, and first differences of A101096.
For other sequences based upon MagicNKZ(n,k,z):
...... |  n = 1  |  n = 2  |  n = 3  |  n = 4  |  n = 5  |  n = 6  |  n = 7  |  n = 8
--------------------------------------------------------------------------------------
z =  0 | A000007 | A019590 | .......  MagicNKZ(n,k,0) = T(n,k+1) from A008292  .......
z =  1 | A000012 | A040000 | A101101 | A101104 | A101100 | ....... | ....... | .......
z =  2 | A000027 | A005408 | A008458 | A101103 | thisSeq | ....... | ....... | .......
z =  3 | A000217 | A000290 | A003215 | A005914 | A101096 | ....... | ....... | .......
z =  4 | A000292 | A000330 | A000578 | A005917 | A101098 | ....... | ....... | .......
z =  5 | A000332 | A002415 | A000537 | A000583 | A022521 | ....... | A255181 | .......
z =  6 | A000389 | A005585 | A024166 | A000538 | A000584 | A022522 | A255177 | A255182
z =  7 | A000579 | A040977 | A101094 | A101089 | A000539 | A001014 | A022523 | A255178
z =  8 | A000580 | A050486 | A101097 | A101090 | A101092 | A000540 | A001015 | A022524
z =  9 | A000581 | A053347 | A101102 | A101091 | A101099 | A101093 | A000541 | A001016
z = 10 | A000582 | A054333 | A254469 | A254681 | A254644 | A254640 | A250212 | A000542
z = 11 | A001287 | A054334 | A254869 | A254470 | A254682 | A254645 | A254641 | A253636
z = 12 | A001288 | A057788 | ....... | A254870 | A254471 | A254683 | A254646 | A254642
z = 13 | A010965 | ....... | ....... | ....... | A254871 | A254472 | A254684 | A254647
z = 14 | A010966 | ....... | ....... | ....... | ....... | A254872 | ....... | .......
--------------------------------------------------------------------------------------
Cf. A047969.

I:=[1,28,121,240,360]; [n le 5 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, May 07 2015

MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}];Table[MagicNKZ, {n, 5, 5}, {z, 2, 2}, {k, 0, 34}]
CoefficientList[Series[(1 + 26 x + 66 x^2 + 26 x^3 + x^4)/(1 - x)^2, {x, 0, 50}], x] (* Vincenzo Librandi, May 07 2015 *)
Join[{1,28,121,240},Differences[Range[50]^5,4]] (* or *) LinearRecurrence[{2,-1},{1,28,121,240,360},50] (* Harvey P. Dale, Jun 11 2016 *)

a(n)=if(n>3, 120*n-240, 33*n^2-72*n+40) \\ Charles R Greathouse IV, Oct 11 2015

[1,28,121]+[120*(k-2) for k in range(4,36)] # Danny Rorabaugh, Apr 23 2015

a(k+1) = Sum_{j=0..k+1} (-1)^j*binomial(n + 1 - z, j)*(k - j + 1)^n; n = 5, z = 2.
For k>3, a(k) = Sum_{j=0..4} (-1)^j*binomial(4, j)*(k - j)^5 = 120*(k - 2).
a(n) = 2*a(n-1) - a(n-2), n>5. G.f.: x*(1+26*x+66*x^2+26*x^3+x^4) / (1-x)^2. - Colin Barker, Mar 01 2012

MagicNKZ material edited, Crossrefs table added, SeriesAtLevelR material removed by Danny Rorabaugh, Apr 23 2015

Name changed and keyword 'uned' removed by Danny Rorabaugh, May 06 2015

A018213 Alkane (or paraffin) numbers l(12,n).

Original entry on oeis.org

1, 5, 30, 110, 365, 1001, 2520, 5720, 12190, 24310, 46252, 83980, 147070, 248710, 408760, 653752, 1021735, 1562275, 2343770, 3453450, 5008003, 7153575, 10080720, 14024400, 19284460, 26225628, 35304920, 47071640

Offset: 0

N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)

Equals (1/2) * ((A000582) + (A000332 interleaved with zeros)) = (1/2) * ((1, 10, 55, 220, 715...) + (1, 0, 5, 0, 15,...)); where A000582 = binomial(n,9) and A000332 = binomial(n,4).

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
Winston C. Yang (paper in preparation).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Classic Sequences
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (5, -5, -15, 35, 1, -65, 45, 45, -65, 1, 35, -15, -5, 5, -1).

[(1/(2*Factorial(9)))*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8)*(n+9)+(1/6)*(1/2^7)*(n+2)*(n+4)*(n+6)*(n+8)*(1/2)*(1+(-1)^n): n in [0..40]]; // Vincenzo Librandi, Oct 16 2013

CoefficientList[Series[(5 x^4 + 10 x^2 + 1)/((x - 1)^10 (x + 1)^5), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 16 2013 *)
LinearRecurrence[{5, -5, -15, 35, 1, -65, 45, 45, -65, 1, 35, -15, -5, 5, -1},{1, 5, 30, 110, 365, 1001, 2520, 5720, 12190, 24310, 46252, 83980, 147070, 248710, 408760},101] (* Ray Chandler, Sep 23 2015 *)

l(c, r) = 1/2 binomial(c+r-3, r) + 1/2 d(c, r), where d(c, r) is binomial((c + r - 3)/2, r/2) if c is odd and r is even, 0 if c is even and r is odd, binomial((c + r - 4)/2, r/2) if c is even and r is even, binomial((c + r - 4)/2, (r - 1)/2) if c is odd and r is odd.
G.f.: (5*x^4+10*x^2+1)/((x-1)^10*(x+1)^5). [Colin Barker, Aug 06 2012]
a(n) = (1/(2*9!))*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8)*(n+9) +(1/6)*(1/2^7)*(n+2)*(n+4)*(n+6)*(n+8)*(1/2)*(1+(-1)^n). [Yosu Yurramendi, Jun 23 2013]

A001781 Expansion of 1/((1+x)*(1-x)^10).

Original entry on oeis.org

1, 9, 46, 174, 541, 1461, 3544, 7896, 16414, 32206, 60172, 107788, 186142, 311278, 505912, 801592, 1241383, 1883167, 2803658, 4103242, 5911763, 8395387, 11764688, 16284112, 22282988, 30168268, 40439192, 53704088, 70699532, 92312108, 119603024, 153835856, 196507709, 249384101

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (9,-35,75,-90,42,42,-90,75,-35,9,-1).

Cf. A000582.

Tenth column of A112465.

[1/2903040*(2*n+11) *(2*n^8 +88*n^7 +1616*n^6 +16060*n^5 +93656*n^4 +324808*n^3 +646236*n^2 +663894*n +263655)+(-1)^n/1024  : n in [0..30]]; // Vincenzo Librandi, Oct 08 2011

R:=PowerSeriesRing(Integers(), 50);
Coefficients(R!( 1/((1+x)*(1-x)^10) )); // G. C. Greubel, Apr 20 2025

A001781 := proc(n) 1/2903040*(2*n+11) *(2*n^8 +88*n^7 +1616*n^6 +16060*n^5 +93656*n^4 +324808*n^3 +646236*n^2 +663894*n +263655)+(-1)^n/1024 ; end proc:
seq(A001781(n),n=0..50) ; # R. J. Mathar, Mar 22 2011

Vec(1/(1+x)/(1-x)^10+O(x^99)) \\ Charles R Greathouse IV, Apr 18 2012

def A001781_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1+x)*(1-x)^10) ).list()
print(A001781_list(50)) # G. C. Greubel, Apr 20 2025

a(n) = +9*a(n-1) -35*a(n-2) +75*a(n-3) -90*a(n-4) +42*a(n-5) +42*a(n-6) -90*a(n-7) +75*a(n-8) -35*a(n-9) +9*a(n-10) -a(n-11). - R. J. Mathar, Mar 22 2011

a(n) + a(n+1) = A000582(n+10). - R. J. Mathar, Jan 06 2021

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)

A053138 Binomial coefficients C(2*n+9,9).

Original entry on oeis.org

1, 55, 715, 5005, 24310, 92378, 293930, 817190, 2042975, 4686825, 10015005, 20160075, 38567100, 70607460, 124403620, 211915132, 350343565, 563921995, 886163135, 1362649145, 2054455634, 3042312350, 4431613550, 6358402050, 8996462475, 12565671261, 17341763505

Offset: 0

Wolfdieter Lang

Even-indexed members of tenth column of Pascal's triangle A007318.

Number of standard tableaux of shape (2n+1,1^9). - Emeric Deutsch, May 30 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Milan Janjić, Two Enumerative Functions.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45, 10,-1).

Cf. A000582, A007318, A053131, A053137.

[Binomial(2*n+9,9): n in [0..30]]; // Vincenzo Librandi, Oct 07 2011

A053138:=n->binomial(2*n+9,9): seq(A053138(n), n=0..40); # Wesley Ivan Hurt, Dec 05 2016

Table[Binomial[2n+9,9],{n,0,30}] (* Harvey P. Dale, Dec 08 2011 *)

for(n=0,30, print1(binomial(2*n+9, 9), ", ")) \\ G. C. Greubel, Sep 03 2018

a(n) = binomial(2*n+9, 9) = A000582(2*n+9).
G.f.: (1 + 45*x + 210*x^2 + 210*x^3 + 45*x^4 + x^5) / (1-x)^10.
G.f.: (1 + x) * (x^4 + 44*x^3 + 166*x^2 + 44*x + 1) / (1-x)^10.
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n > 9. - Wesley Ivan Hurt, Dec 05 2016
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=0} 1/a(n) = 1152*log(2) - 27912/35.
Sum_{n>=0} (-1)^n/a(n) = 36*Pi - 3924/35. (End)

A095704 Triangle read by rows giving coefficients of the trigonometric expansion of sin(n*x).

Original entry on oeis.org

1, 2, 0, 3, 0, -1, 4, 0, -4, 0, 5, 0, -10, 0, 1, 6, 0, -20, 0, 6, 0, 7, 0, -35, 0, 21, 0, -1, 8, 0, -56, 0, 56, 0, -8, 0, 9, 0, -84, 0, 126, 0, -36, 0, 1, 10, 0, -120, 0, 252, 0, -120, 0, 10, 0, 11, 0, -165, 0, 462, 0, -330, 0, 55, 0, -1, 12, 0, -220, 0, 792, 0, -792, 0, 220, 0, -12, 0, 13, 0, -286, 0, 1287, 0

Offset: 1

Robert G. Wilson v, Jul 06 2004

			The trigonometric expansion of sin(4x) is 4*cos(x)^3*sin(x) - 4*cos(x)*sin(x)^3, so the fourth row is 4, 0, -4, 0.
Triangle begins:
1
2 0
3 0 -1
4 0 -4 0
5 0 -10 0 1
6 0 -20 0 6 0
7 0 -35 0 21 0 -1
8 0 -56 0 56 0 -8 0

Clark Kimberling, Polynomials associated with reciprocation, JIS 12 (2009) 09.3.4, section 5.

First column is A000027 = C(n, 1), third column is A000292 = C(n, 3), fifth column is A000389 = C(n, 5), seventh column is A000580 = C(n, 7), ninth column is A000582 = C(n, 9).
A001288 = C(n, 11), A010966 = C(n, 13), A010968 = C(n, 15), A010970 = C(n, 17), A010972 = C(n, 19),
A010974 = C(n, 21), A010976 = C(n, 23), A010978 = C(n, 25), A010980 = C(n, 27), A010982 = C(n, 29),
A010984 = C(n, 31), A010986 = C(n, 33), A010988 = C(n, 35), A010990 = C(n, 37), A010992 = C(n, 39),
A010994 = C(n, 41), A010996 = C(n, 43), A010998 = C(n, 45), A011000 = C(n, 47), A017713 = C(n, 49)
Another version of the triangle in A034867. Cf. A096754.
A017715 = C(n, 51), A017717 = C(n, 53), A017719 = C(n, 55), A017721 = C(n, 57), etc.

Flatten[ Table[ Plus @@ CoefficientList[ TrigExpand[ Sin[n*x]], {Sin[x], Cos[x]}], {n, 13}]]

T(n,k) = C(n+1,k+1)*sin(Pi*(k+1)/2). - Paul Barry, May 21 2006

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

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

A095665 Tenth column (m=9) of (1,3)-Pascal triangle A095660.

Original entry on oeis.org

3, 28, 145, 550, 1705, 4576, 11011, 24310, 50050, 97240, 179894, 319124, 545870, 904400, 1456730, 2288132, 3513917, 5287700, 7811375, 11347050, 16231215, 22891440, 31865925, 43826250, 59603700, 80219568, 106919868, 141214920

Offset: 0

Wolfdieter Lang, Jun 11 2004

If Y is a 3-subset of an n-set X then, for n>=11, a(n-11) is the number of 9-subsets of X having at most one element in common with Y. - Milan Janjic, Nov 23 2007

Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

Ninth column: A095663.

Table[Binomial[n+8,8] (n+27)/9,{n,0,30}] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{3,28,145,550,1705,4576,11011,24310,50050,97240},30] (* Harvey P. Dale, Oct 13 2017 *)

a(n)= binomial(n+8, 8)*(n+27)/9 = 3*b(n)-2*b(n-1), with b(n):=binomial(n+9, 9); cf. A000582.

G.f.: (3-2*x)/(1-x)^10.

cp's OEIS Frontend

A097300 Tenth column (m=9) of (1,6)-Pascal triangle A096956.

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A101095 Fourth difference of fifth powers (A000584).

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A018213 Alkane (or paraffin) numbers l(12,n).

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A001781 Expansion of 1/((1+x)*(1-x)^10).

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

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A053138 Binomial coefficients C(2*n+9,9).

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A095704 Triangle read by rows giving coefficients of the trigonometric expansion of sin(n*x).

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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.