A000580 -id:A000580 - cp's OEIS frontend

A039948 A triangle related to A000045 (Fibonacci numbers).

1, 1, 1, 4, 2, 1, 18, 12, 3, 1, 120, 72, 24, 4, 1, 960, 600, 180, 40, 5, 1, 9360, 5760, 1800, 360, 60, 6, 1, 105840, 65520, 20160, 4200, 630, 84, 7, 1, 1370880, 846720, 262080, 53760, 8400, 1008, 112, 8, 1, 19958400, 12337920, 3810240, 786240, 120960, 15120, 1512, 144, 9, 1

Offset: 0

Wolfdieter Lang

			Triangle begins :
    1;
    1,   1;
    4,   2,   1;
   18,  12,   3,  1;
  120,  72,  24,  4, 1;
  960, 600, 180, 40, 5, 1;
... - _Philippe Deléham_, Nov 08 2011

Seiichi Manyama, Rows n = 0..139, flattened
P. R. J. Asveld, Fibonacci-like differential equations with a polynomial nonhomogeneous term, Fib. Quart. 27 (1989), 303-309.

Cf. A000045, A000142, A005442, A005443, A110313 (row sums).

Diagonals include: A000027, A000217, A000292, A000332, A000389, A000579, A000580, A000581, A000582, A001287, A001288, A010965, A010966.

[(Factorial(n)/Factorial(k))*Fibonacci(n-k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2022

T[n_,k_]:= (n!/k!)*Fibonacci[n-k+1];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2022 *)

def A039948(n, k): return factorial(n-k)*binomial(n,k)*fibonacci(n-k+1)
flatten([[A039948(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 20 2022

T(n, m) = n!*Fibonacci(n-m+1)/m!, n >= m >= 0.
T(n, 0) = A005442(n).
T(n, 1) = A005443(n).
E.g.f. for column m: x^m/(m!*(1-x-x^2)), m >= 0.
From G. C. Greubel, Nov 20 2022: (Start)
T(n, n-1) = A000027(n).
T(n, n-2) = 4*A000217(n-1), n >= 2.
T(n, n-3) = 18*A000292(n-2), n >= 3.
T(n, n-4) = 5! * A000332(n), n >= 4.
T(n, n-5) = 8 * 5! * A000389(n), n >= 5.
T(n, n-6) = 13 * 6! * A000579(n), n >= 6.
T(n, n-7) = 21 * 7! * A000580(n), n >= 7.
T(n, n-8) = 34 * 8! * A000581(n), n >= 8.
T(n, n-9) = 55 * 9! * A000582(n), n >= 9.
T(n, n-10) = 89 * 10! * A001287(n), n >= 10.
T(n, n-11) = 12 * 12! * A001288(n), n >= 11.
T(n, n-12) = 233 * 12! * A010965(n), n >= 12.
T(n, n-13) = 89 * 13! * A010966(n), n >= 13.
Sum_{k=0..n} T(n, k) = A110313(n). (End)

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

A097298 Eighth column (m=7) of (1,6)-Pascal triangle A096956.

Original entry on oeis.org

6, 43, 176, 540, 1380, 3102, 6336, 12012, 21450, 36465, 59488, 93704, 143208, 213180, 310080, 441864, 618222, 850839, 1153680, 1543300, 2039180, 2664090, 3444480, 4410900, 5598450, 7047261, 8803008, 10917456, 13449040, 16463480

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 (8,-28,56,-70,56,-28,8,-1).

Cf. A000580, A096956.

Cf. other columns: A096957 (m = 3), A096958 (m = 4), A096959 (m = 5), A097297 (m = 6), A097299 (m = 8), A097300 (m = 9).

A097298[n_] := (n + 42)*Binomial[n + 6, 6]/7;
Array[A097298, 50, 0] (* Paolo Xausa, May 02 2025 *)

a(n) = A096956(n+7, 7) = 6*b(n) - 5*b(n-1) = (n+42)*binomial(n+6, 6)/7, with b(n) = A000580(n+7) = binomial(n+7, 7).

G.f.: (6-5*x)/(1-x)^8.

A323535 a(n) = Product_{k=1..n} (binomial(k-1,7) + binomial(n-k,7)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 240248274716697412239360000, 5659588189073370681080838881280000, 148305406398618918682372310424354816000000, 4049882681498254991937037064898924144230400000000, 137651993399006086593846978063252515678682995490816000000

Offset: 0

Vaclav Kotesovec, Jan 17 2019

nonn

Cf. A000580, A323425, A323496, A323497, A323533, A323534.

Table[Product[Binomial[k-1,7] + Binomial[n-k,7], {k, 1, n}], {n, 0, 20}]

a(n) = prod(k=1, n, binomial(k-1, 7) + binomial(n-k, 7)); \\ Daniel Suteu, Jan 17 2019

a(n) ~ exp(-7*n + (n-7)*(1 + c*Pi)) * n^(7*n) / (7!)^n, where c = 8*cos((Pi + arctan(2769*sqrt(3)/239))/6) / sqrt(21) = 1.2446281707164555154936427017... is the root of the equation 823543*c^6 - 3764768*c^4 + 4302592*c^2 - 692224 = 0.

A018211 Alkane (or paraffin) numbers l(10,n).

Original entry on oeis.org

1, 4, 20, 60, 170, 396, 868, 1716, 3235, 5720, 9752, 15912, 25236, 38760, 58200, 85272, 122661, 173052, 240460, 328900, 444158, 592020, 780572, 1017900, 1315015, 1682928, 2136304, 2689808, 3362600, 4173840, 5148144, 6310128

Offset: 0

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

nonn

Equals (1/2) * ((1, 8, 36, 120, 330, 792,...) + (1, 0, 4, 0, 10, 0, 20,...)); where (1, 8, 36,..) = A000580 = C(n,7), and (1, 4, 10,...) = the Tetrahedral numbers.

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

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 (4, -2, -12, 17, 8, -28, 8, 17, -12, -2, 4, -1).

Cf. A282011.

a:= n-> (Matrix([[1, 0$7, -1, -4, -20, -60]]). Matrix(12, (i,j)-> `if`(i=j-1, 1, `if`(j=1, [4, -2, -12, 17, 8, -28, 8, 17, -12, -2, 4, -1][i], 0)))^n)[1,1]: seq(a(n), n=0..31); # Alois P. Heinz, Jul 31 2008

LinearRecurrence[{4, -2, -12, 17, 8, -28, 8, 17, -12, -2, 4, -1},{1, 4, 20, 60, 170, 396, 868, 1716, 3235, 5720, 9752, 15912},32] (* Ray Chandler, Sep 23 2015 *)

G.f.: (1+6*x^2+x^4)/((1-x)^4*(1-x^2)^4). [ N. J. A. Sloane ]
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.
a(n) = (1/(2*7!))*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7) + (1/3)*(1/2^5)*(n+2)*(n+4)*(n+6)*(1/2)*(1+(-1)^n) [Yosu Yurramendi Jun 23 2013]

A053136 Binomial coefficients C(2*n+7,7).

Original entry on oeis.org

1, 36, 330, 1716, 6435, 19448, 50388, 116280, 245157, 480700, 888030, 1560780, 2629575, 4272048, 6724520, 10295472, 15380937, 22481940, 32224114, 45379620, 62891499, 85900584, 115775100, 154143080, 202927725, 264385836, 341149446, 436270780, 553270671, 696190560

Offset: 0

Wolfdieter Lang

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

Number of standard tableaux of shape (2n+1,1^7). - 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 (8,-28,56,-70,56,-28,8,-1).

Cf. A007318, A053135, A000580, A053129.

[Binomial(2*n+7,7): n in [0..30]]; // Vincenzo Librandi, Oct 07 2011

Table[Binomial[2*n+7, 7], {n, 0, 30}] (* G. C. Greubel, Sep 03 2018 *)

a(n)=binomial(2*n+7,7) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = binomial(2*n+7, 7) = A000580(2*n+7).
G.f.: (1 + 28*x + 70*x^2 + 28*x^3 + x^4)/(1-x)^8.
E.g.f.: (630 + 22050*x + 81585*x^2 + 87465*x^3 + 36960*x^4 + 6888*x^5 + 560*x^6 + 16*x^7)*exp(x)/630. - G. C. Greubel, Sep 03 2018
From Amiram Eldar, Nov 03 2022: (Start)
Sum_{n>=0} 1/a(n) = 224*log(2) - 4627/30.
Sum_{n>=0} (-1)^n/a(n) = 28*log(2) - 553/30. (End)

A092056 Square table read by downward antidiagonals where T(n,k) = binomial(n+2^k-1,n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 10, 4, 1, 1, 16, 36, 20, 5, 1, 1, 32, 136, 120, 35, 6, 1, 1, 64, 528, 816, 330, 56, 7, 1, 1, 128, 2080, 5984, 3876, 792, 84, 8, 1, 1, 256, 8256, 45760, 52360, 15504, 1716, 120, 9, 1, 1, 512, 32896, 357760, 766480, 376992, 54264, 3432, 165, 10, 1

Offset: 0

Henry Bottomley, Feb 19 2004

Each column is convolution of preceding column starting from the all 1's sequence.

T(n,k) is the number of relations between a set of k distinguishable elements and a set of n indistinguishable elements. - Isaac R. Browne, May 14 2025

			Rows start:
  1, 1,  1,   1,    1,     1,      1,...
  1, 2,  4,   8,   16,    32,     64,...
  1, 3, 10,  36,  136,   528,   2080,...
  1, 4, 20, 120,  816,  5984,  45760,...
  1, 5, 35, 330, 3876, 52360, 766480,...
  ...

Columns include (essentially) A000012, A000027, A000292, A000580, A010968, etc.
Rows include A000012, A000079, A007582, A092056.
Main diagonal gives A060690.
Cf. A137153 (same with reflected antidiagonals).

T(n,k) = Sum_{i=0..n} T(i,k-1)*T(n-i,k-1) starting with T(n,0) = 1 for n>=0.

A133084 A007318 * A133080.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 4, 1, 5, 4, 10, 4, 1, 6, 5, 20, 10, 6, 1, 7, 6, 35, 20, 21, 6, 1, 8, 7, 56, 35, 56, 21, 8, 1, 9, 8, 84, 56, 126, 56, 36, 8, 1, 10, 9, 120, 84, 252, 126, 120, 36, 10, 1, 11, 10, 165, 120, 462, 252, 330, 120, 55, 10, 1

Offset: 1

Gary W. Adamson, Sep 16 2007

Row sums = A003945: (1, 3, 6, 12, 24, 48, 96, ...).

A133084 is jointly generated with A133567 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=x*u(n-1,x)+v(n-1,x)+1. See the Mathematica section. - Clark Kimberling, Feb 28 2012

			First few rows of the triangle:
  1;
  2,  1;
  3,  2,  1;
  4,  3,  4,  1;
  5,  4, 10,  4,  1;
  6,  5, 20, 10,  6,  1;
  7,  6, 35, 20, 21,  6,  1;
  ...

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

Cf. A133080, A003945, A133567.

Cf. A000292 (column 3 and 4), A000389 (column 5 and 6), A000580 (column 7).

/* As triangle */ [[(1-(1+(-1)^k)/2 )*Binomial(n, k)+((1+(-1)^k)/2)*Binomial(n-1, k-1): k in [1..n]]: n in [1.. 11]]; // Vincenzo Librandi, Oct 21 2017

A133084 := proc(n,k)
    add(binomial(n-1,i-1)*A133080(i,k),i=1..n) ;
end proc: # R. J. Mathar, Jun 13 2025

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A133567 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A133084 *)
(* Clark Kimberling, Feb 28 2012 *)
T[n_, k_] := If[k == n, 1, (1  - (1 + (-1)^k)/2 )*Binomial[n, k] + ((1 + (-1)^k)/2)*Binomial[n - 1, k - 1]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* G. C. Greubel, Oct 21 2017 *)

for(n=1,10, for(k=1,n, print1(if(k == n, 1, (1  - (1 + (-1)^k)/2 )*binomial(n, k) + ((1 + (-1)^k)/2)*binomial(n - 1, k - 1)), ", "))) \\ G. C. Greubel, Oct 21 2017

Binomial transform of triangle A133080.

A095663 Eighth column (m=7) of (1,3)-Pascal triangle A095660.

Original entry on oeis.org

3, 22, 92, 288, 750, 1716, 3564, 6864, 12441, 21450, 35464, 56576, 87516, 131784, 193800, 279072, 394383, 547998, 749892, 1012000, 1348490, 1776060, 2314260, 2985840, 3817125, 4838418, 6084432, 7594752, 9414328, 11594000, 14191056

Offset: 0

Wolfdieter Lang, Jun 11 2004

If Y is a 3-subset of an n-set X then, for n>=9, a(n-9) is the number of 7-subsets of X having at most one element in common with Y. - Milan Janjic, Nov 23 2007

Seventh column: A095662. Ninth column: A095664.

G.f.: (3-2*x)/(1-x)^8.

a(n)= binomial(n+6, 6)*(n+21)/7 = 3*b(n)-2*b(n-1), with b(n):=binomial(n+7, 7); cf. A000580.

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

cp's OEIS Frontend

A039948 A triangle related to A000045 (Fibonacci numbers).

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A101095 Fourth difference of fifth powers (A000584).

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A097298 Eighth column (m=7) of (1,6)-Pascal triangle A096956.

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A323535 a(n) = Product_{k=1..n} (binomial(k-1,7) + binomial(n-k,7)).

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Mathematica

PARI

Formula

A018211 Alkane (or paraffin) numbers l(10,n).

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Maple

Mathematica

Formula

A053136 Binomial coefficients C(2*n+7,7).

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Magma

Mathematica

PARI

Formula

A092056 Square table read by downward antidiagonals where T(n,k) = binomial(n+2^k-1,n).

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Formula

A133084 A007318 * A133080.

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