A001789 -id:A001789 - cp's OEIS frontend

A178820 Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.

1, 4, 4, 10, 20, 10, 20, 60, 60, 20, 35, 140, 210, 140, 35, 56, 280, 560, 560, 280, 56, 84, 504, 1260, 1680, 1260, 504, 84, 120, 840, 2520, 4200, 4200, 2520, 840, 120, 165, 1320, 4620, 9240, 11550, 9240, 4620, 1320, 165, 220, 1980, 7920, 18480, 27720, 27720, 18480, 7920, 1980, 220

Offset: 0

Harlan J. Brothers, Jun 17 2010

The product of the tetrahedral numbers (A000292, beginning with second term) and Pascal's triangle (A007318). Also level 4 of Pascal's prism (A178819): (i+3; 3, i-j, j), i >= 0, 0 <= j <= i.

			Triangle begins:
   1;
   4,   4;
  10,  20,  10;
  20,  60,  60,  20;
  35, 140, 210, 140,  35;

G. C. Greubel, Rows n=0..100 of triangle, flattened
H. J. Brothers, Pascal's prism, The Mathematical Gazette, 96 (July 2012), 213-220.
H. J. Brothers, Pascal's Prism: Supplementary Material

Cf. A000292, A007318, A178819.

Rows sums give A001789.

T:=Flat(List([0..10], n-> List([0..n], k-> Binomial(n+3, 3)* Binomial(n, k) ))); # G. C. Greubel, Jan 22 2019

/* As triangle */ [[Binomial(n+3,3)*Binomial(n,k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Oct 23 2017

T:=(n,k)->binomial(n+3,3)*binomial(n,k): seq(seq(T(n,k),k=0..n),n=0..9); # Muniru A Asiru, Jan 22 2019

Table[Multinomial[3, i-j, j], {i, 0, 9}, {j, 0, i}]//Column

{T(n,k) = binomial(n+3, 3)*binomial(n, k)}; \\ G. C. Greubel, Jan 22 2019

[[binomial(n+3, 3)*binomial(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 22 2019

T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
For element a in A178819: a_(4, i, j) = (i+2; 3, i-j, j-1), i >= 1, 1 <= j <= i.
G.f.: 1/(1 - x - x*y)^4. - Ilya Gutkovskiy, Mar 20 2020

A130812 If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 6-subsets of X containing none of X_i, (i=1,...n).

Original entry on oeis.org

64, 448, 1792, 5376, 13440, 29568, 59136, 109824, 192192, 320320, 512512, 792064, 1188096, 1736448, 2480640, 3472896, 4775232, 6460608, 8614144, 11334400, 14734720, 18944640, 24111360, 30401280, 38001600, 47121984, 57996288, 70884352, 86073856, 103882240

Offset: 6

Milan Janjic, Jul 16 2007

Number of n permutations (n>=6) of 3 objects u,v,z, with repetition allowed, containing n-6 u's. Example: if n=6 then n-6 =(0) zero u, a(1)=64. - Zerinvary Lajos, Aug 05 2008

a(n) is the number of 5-dimensional elements in an n-cross polytope where n>=6. - Patrick J. McNab, Jul 06 2015

Vincenzo Librandi, Table of n, a(n) for n = 6..1000
Milan Janjic, Two Enumerative Functions
Eric Weisstein's World of Mathematics, Cross Polytope

Cf. A038207, A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354, A046092, A130809, A130810, A130811. - Zerinvary Lajos, Aug 05 2008

[Binomial(2*n,6)+Binomial(n,2)*Binomial(2*n-4,2)- n*Binomial(2*n-2,4)-Binomial(n,3): n in [6..40]]; // Vincenzo Librandi, Jul 09 2015

a:=n->binomial(2*n,6)+binomial(n,2)*binomial(2*n-4,2)-n*binomial(2*n-2,4)-binomial(n,3);
seq(binomial(n,n-6)*2^6,n=6..32); # Zerinvary Lajos, Dec 07 2007
seq(binomial(n+5, 6)*2^6, n=1..22); # Zerinvary Lajos, Aug 05 2008

CoefficientList[Series[64/(1-x)^7,{x,0,30}],x] (* Vincenzo Librandi, Mar 21 2012 *)

a(n) = binomial(2*n,6) + binomial(n,2)*binomial(2*n-4,2) - n*binomial(2*n-2,4) - binomial(n,3).
a(n) = C(n,n-6)*2^6, n>=6. - Zerinvary Lajos, Dec 07 2007
G.f.: 64*x^6/(1-x)^7. - Colin Barker, Mar 20 2012

A172242 Number of 10-D hypercubes in an n-dimensional hypercube.

Original entry on oeis.org

1, 22, 264, 2288, 16016, 96096, 512512, 2489344, 11202048, 47297536, 189190144, 722362368, 2648662016, 9372188672, 32133218304, 107110727680, 348109864960, 1105760747520, 3440144547840, 10501493882880, 31504481648640

Offset: 10

Zerinvary Lajos, Jan 29 2010

With a different offset, number of n-permutations (n>=8) of 3 objects: u, v, z with repetition allowed, containing exactly ten (10) u's.

Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (22,-220,1320,-5280,14784,-29568,42240,-42240,28160,-11264,2048).

Cf. A001788, A001789, A003472, A038207, A054849, A002409, A140325, A054851, A140354.

Table[Binomial[n + 10, 10]*2^n, {n, 0, 22}]

[lucas_number2(n, 2, 0)*binomial(n,10)/2^10 for n in range(10, 31)] # Zerinvary Lajos, Feb 05 2010

a(n) = A038207(n,10).
a(n) = binomial(n,10)*2^(n-10). [Corrected by R. J. Mathar, Feb 21 2010]
G.f.: -x^10/(2*x-1)^11. - Colin Barker, Nov 11 2012
a(n) = Sum_{i=10..n} binomial(i,10)*binomial(n,i). Example: for n=15, a(15) = 1*3003 + 11*1365 + 66*455 + 286*105 + 1001*15 + 3003*1 = 96096. - Bruno Berselli, Mar 23 2018
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=10} 1/a(n) = 1879/126 - 20*log(2).
Sum_{n>=10} (-1)^n/a(n) = 393660*log(3/2) - 20111419/126. (End)

A130811 If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 5-subsets of X containing none of X_i, (i=1,...n).

Original entry on oeis.org

32, 192, 672, 1792, 4032, 8064, 14784, 25344, 41184, 64064, 96096, 139776, 198016, 274176, 372096, 496128, 651168, 842688, 1076768, 1360128, 1700160, 2104960, 2583360, 3144960, 3800160, 4560192, 5437152, 6444032, 7594752, 8904192

Offset: 5

Milan Janjic, Jul 16 2007

Number of n permutations (n>=5) of 3 objects u,v,z, with repetition allowed, containing n-5 u's. Example: if n=5 then n-5 =(0) zero u, a(1)=32. - Zerinvary Lajos, Aug 05 2008

a(n) is the number of 4-dimensional elements in an n-cross polytope where n>=5. - Patrick J. McNab, Jul 06 2015

Milan Janjic, Two Enumerative Functions
Eric Weisstein's World of Mathematics, Cross Polytope

Cf. A038207, A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354, A046092, A130809, A130810. - Zerinvary Lajos, Aug 05 2008

[Binomial(n,n-5)*2^5: n in [5..40]]; // Vincenzo Librandi, Jul 09 2015

a:=n->binomial(2*n,5)+(2*n-4)*binomial(n,2)-n*binomial(2*n-2,3)
seq(binomial(n,n-5)*2^5,n=5..34); # Zerinvary Lajos, Dec 07 2007
seq(binomial(n+4, 5)*2^5, n=1..22); # Zerinvary Lajos, Aug 05 2008

Table[Binomial[2 n, 5] + (2 n - 4) Binomial[n, 2] - n Binomial[2 n - 2, 3], {n, 5, 40}] (* Vincenzo Librandi, Jul 09 2015 *)

a(n) = binomial(2*n,5) + (2*n-4)*binomial(n,2) - n*binomial(2*n-2,3).
a(n) = C(n,n-5)*2^5, for n>=5. - Zerinvary Lajos, Dec 07 2007
G.f.: 32*x^5/(1-x)^6. - Colin Barker, Apr 14 2012

A213345 3-quantum transitions in systems of N>=3 spin 1/2 particles, in columns by combination indices.

Original entry on oeis.org

1, 8, 40, 5, 160, 60, 560, 420, 21, 1792, 2240, 336, 5376, 10080, 3024, 84, 15360, 40320, 20160, 1680, 42240, 147840, 110880, 18480, 330, 112640, 506880, 532224, 147840, 7920, 292864, 1647360, 2306304, 960960, 102960

Offset: 3

Stanislav Sykora, Jun 12 2012

For a general discussion, please see A213343.
This a(n) is for triple-quantum transitions (q = 3).
It lists the flattened triangle T(3;N,k) with rows N = 3,5,... and columns k = 0..floor((N-3)/2).

			Some of the 40 triple-quantum transitions for N = 5 and combination index 0: (00000,01011),(10010,11111),...
Starting rows of the triangle T(3;N,k):
  N | k = 0, 1, ..., floor((N-3)/2)
  3 |   1
  4 |   8
  5 |  40   5
  6 | 160  60
  7 | 560 420 21

See A213343

Stanislav Sykora, Table of n, a(n) for n = 3..2452
Stanislav Sykora, T(3;N,k) with rows N=3,..,100 and columns k=0,..,floor((N-3)/2)
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

Cf. A051288 (q=0), A213343 (q=1), A213344 (q=2), A213346 to A213352 (q=4..10).

Cf. A001789 (first column), A002696 (row sums).

With[{q = 3}, Table[2^(n - q - 2 k)*Binomial[n, k] Binomial[n - k, q + k], {n, 13}, {k, 0, Floor[(n - q)/2]}]] // Flatten (* Michael De Vlieger, Nov 18 2019 *)

PARI
```
See A213343; set thisq = 3.
```

Set q = 3 in: T(q;N,k) = 2^(N-q-2*k)*binomial(N,k)*binomial(N-k,q+k).

A248826 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x+k)^k for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, -3, 1, 0, 6, -8, 1, 0, -10, 40, -15, 1, 0, 15, -160, 135, -24, 1, 0, -21, 560, -945, 336, -35, 1, 0, 28, -1792, 5670, -3584, 700, -48, 1, 0, -36, 5376, -30618, 32256, -10500, 1296, -63, 1, 0, 45, -15360, 153090, -258048, 131250, -25920, 2205, -80, 1, 0, -55, 42240, -721710, 1892352, -1443750, 427680, -56595, 3520, -99, 1

Offset: 0

Derek Orr, Oct 15 2014

Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x+0)^0 + A_1*(x+1)^1 + A_2*(x+2)^2 + ... + A_n*(x+n)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

			1;
0,   1;
0,  -3,      1;
0,   6,     -8,       1;
0, -10,     40,     -15,       1;
0,  15,   -160,     135,     -24,        1;
0, -21,    560,    -945,     336,      -35,      1;
0,  28,  -1792,    5670,   -3584,      700,    -48,      1;
0, -36,   5376,  -30618,   32256,   -10500,   1296,    -63,    1;
0,  45, -15360,  153090, -258048,   131250, -25920,   2205,  -80,   1;
0, -55,  42240, -721710, 1892352, -1443750, 427680, -56595, 3520, -99, 1;

Cf. A001789, A000217.

for(n=0,20,for(k=0,n,if(!k,if(n,print1(0,", "));if(!n,print1(1,", ")));if(k,print1(-sum(i=1,n,((-k)^(i-k-1)*i*binomial(i,k))),", "))))

T(n,1) = n*(n+1)*(-1)^(n+1)/2 for n > 0.
T(n,2) = Binomial(n+1,3)*2^(n-2)*(-1)^n for n > 1.
T(n,n-1) = 1 - n^2 for n > 0.
T(n,n-2) = (1/2)*n*(n-2)^2*(n+1) for n > 1.

A317495 Triangle read by rows: T(0,0) = 1; T(n,k) =2 * T(n-1,k) + T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 2, 4, 8, 1, 16, 4, 32, 12, 64, 32, 1, 128, 80, 6, 256, 192, 24, 512, 448, 80, 1, 1024, 1024, 240, 8, 2048, 2304, 672, 40, 4096, 5120, 1792, 160, 1, 8192, 11264, 4608, 560, 10, 16384, 24576, 11520, 1792, 60, 32768, 53248, 28160, 5376, 280, 1, 65536, 114688, 67584, 15360, 1120, 12

Offset: 0

Zagros Lalo, Jul 30 2018

The numbers in rows of the triangle are along a "second layer" of skew diagonals pointing top-left in center-justified triangle given in A013609 ((1+2*x)^n) and along a "second layer" of skew diagonals pointing top-right in center-justified triangle given in A038207 ((2+x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (1+2*x)^n and (2+x)^n are given in A128099 and A207538 respectively.)
The coefficients in the expansion of 1/(1-2x-x^3) are given by the sequence generated by the row sums.
The row sums give A008998 and Pisot sequences E(4,9), P(4,9) when n > 1, see A020708.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.205569430400..., when n approaches infinity.

			Triangle begins:
       1;
       2;
       4;
       8,      1;
      16,      4;
      32,     12;
      64,     32,      1;
     128,     80,      6;
     256,    192,     24;
     512,    448,     80,      1;
    1024,   1024,    240,      8;
    2048,   2304,    672,     40;
    4096,   5120,   1792,    160,     1;
    8192,  11264,   4608,    560,    10;
   16384,  24576,  11520,   1792,    60;
   32768,  53248,  28160,   5376,   280,   1;
   65536, 114688,  67584,  15360,  1120,  12;
  131072, 245760, 159744,  42240,  4032,  84;
  262144, 524288, 372736, 112640, 13440, 448, 1;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 358, 359.

Row sums give A008998, A020708.
Cf. A013609
Cf. A038207
Cf. A317494
Cf. A128099, A207538.
Cf. A000079 (column 0), A001787 (column 1), A001788 (column 2), A001789 (column 3), A003472 (column 4).

Flat(List([0..20],n->List([0..Int(n/3)],k->2^(n-3*k)/(Factorial(n-3*k)*Factorial(k))*Factorial(n-2*k)))); # Muniru A Asiru, Jul 31 2018

/* As triangle */ [[2^(n-3*k)/(Factorial(n-3*k)*Factorial(k))* Factorial(n-2*k): k in [0..Floor(n/3)]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 05 2018

t[n_, k_] := t[n, k] = 2^(n - 3k)/((n - 3 k)! k!) (n - 2 k)!; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/3]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 t[n - 1, k] + t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/3]}] // Flatten

T(n,k) = 2^(n - 3k) / ((n - 3k)! k!) * (n - 2k)! where n >= 0 and k = 0..floor(n/3).

A318776 Triangle read by rows: T(0,0) = 1; T(n,k) = 2*T(n-1,k) + T(n-5,k-1) for k = 0..floor(n/5); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 1, 64, 4, 128, 12, 256, 32, 512, 80, 1024, 192, 1, 2048, 448, 6, 4096, 1024, 24, 8192, 2304, 80, 16384, 5120, 240, 32768, 11264, 672, 1, 65536, 24576, 1792, 8, 131072, 53248, 4608, 40, 262144, 114688, 11520, 160, 524288, 245760, 28160, 560, 1048576, 524288, 67584, 1792, 1, 2097152, 1114112, 159744, 5376, 10

Offset: 0

Zagros Lalo, Sep 04 2018

The numbers in rows of the triangle are along a "fourth layer" skew diagonals pointing top-left in center-justified triangle given in A013609 ((1+2*x)^n) and along a "fourth layer" skew diagonals pointing top-right in center-justified triangle given in A038207 ((2+x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (1+2*x)^n and (2+x)^n are given in A128099 and A207538 respectively.)
The coefficients in the expansion of 1/(1-2*x-x^5) are given by the sequence generated by the row sums.
The row sums give A098588.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.0559673967128..., when n approaches infinity.

			Triangle begins:
        1;
        2;
        4;
        8;
       16;
       32,       1;
       64,       4;
      128,      12;
      256,      32;
      512,      80;
     1024,     192,      1;
     2048,     448,      6;
     4096,    1024,     24;
     8192,    2304,     80;
    16384,    5120,    240;
    32768,   11264,    672,    1;
    65536,   24576,   1792,    8;
   131072,   53248,   4608,   40;
   262144,  114688,  11520,  160;
   524288,  245760,  28160,  560;
  1048576,  524288,  67584, 1792,  1;
  2097152, 1114112, 159744, 5376, 10;
  ...

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

Row sums give A098588.
Cf. A013609, A038207, A128099, A207538.
Cf. also A000079 (column 0), A001787 (column 1), A001788 (column 2), A001789 (column 3)

t[n_, k_] := t[n, k] = 2^(n - 5 k)/((n - 5 k)! k!) (n - 4 k)!; Table[t[n, k], {n, 0, 21}, {k, 0, Floor[n/5]} ] // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 t[n - 1, k] + t[n - 5, k - 1]]; Table[t[n, k], {n, 0, 21}, {k, 0, Floor[n/5]}] // Flatten

T(n,k) = 2^(n - 5*k) / ((n - 5*k)! k!) * (n - 4*k)! where n >= 0 and 0 <= k <= floor(n/5).

A359662 Number of (3-dimensional) cells of regular m-polytopes for m >= 3.

Original entry on oeis.org

1, 5, 8, 15, 16, 24, 35, 40, 70, 80, 120, 126, 160, 210, 240, 330, 495, 560, 600, 715, 1001, 1120, 1365, 1792, 1820, 2016, 2380, 3060, 3360, 3876, 4845, 5280, 5376, 5985, 7315, 7920, 8855, 10626, 11440, 12650, 14950, 15360, 16016, 17550, 20475, 21840, 23751

Offset: 1

Marco Ripà, Jan 10 2023

In 3 dimensions there are five (convex) regular polytopes and each of them (trivially) consists of a single cell.
In 4 dimensions there are six regular 4-polytopes and they have 5, 8, 16, 24, 120, 600 3-dimensional cells (A063924).
In m >= 5 dimensions, there are only 3 regular polytopes (i.e., the m-simplex, the m-cube, and the m-crosspolytope) so that we can sort their number of (3-dimensional) cells in ascending order and define the present sequence.

			8 is a term since the hypersurface of a tesseract consists of 8 (cubical) cells.

MathStackExchange, MathStackExchange discussion concerning the number of edges of convex regular n-polytopes
Wikipedia, List of regular polytopes and compounds

Cf. A000332, A001789, A063924, A130810.

Cf. A359201 (edges), A359202 (faces).

Equals {{24, 120, 600} U {A000332} U {A001789} U {A130810}} \ {0}.

A130813 If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 7-subsets of X containing none of X_i, (i=1,...n).

Original entry on oeis.org

128, 1024, 4608, 15360, 42240, 101376, 219648, 439296, 823680, 1464320, 2489344, 4073472, 6449664, 9922560, 14883840, 21829632, 31380096, 44301312, 61529600, 84198400, 113667840, 151557120, 199779840, 260582400, 336585600, 430829568

Offset: 7

Milan Janjic, Jul 16 2007

nonn

Number of n permutations (n>=7) of 3 objects u,v,z, with repetition allowed, containing n-7 u's. Example: if n=7 then n-7 =(0) zero u, a(1)=128. - Zerinvary Lajos, Aug 05 2008

a(n) is the number of 6-dimensional elements in an n-cross polytope where n>=7. - Patrick J. McNab, Jul 06 2015

Milan Janjic, Two Enumerative Functions
Eric Weisstein's World of Mathematics, Cross Polytope

Cf. A038207, A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354, A046092, A130809, A130810, A130811, A130812. - Zerinvary Lajos, Aug 05 2008

Cf. A000580.

[Binomial(n,n-7)*2^7: n in [7..40]]; // Vincenzo Librandi, Jul 09 2015

a:=n->binomial(2*n,7)+binomial(n,2)*binomial(2*n-4,3)-n*binomial(2*n-2,5)-(2*n-6)*binomial(n,3);
seq(binomial(n,n-7)*2^7,n=7..32); # Zerinvary Lajos, Dec 07 2007
seq(binomial(n+6, 7)*2^7, n=1..22); # Zerinvary Lajos, Aug 05 2008

Table[Binomial[n, n - 7] 2^7, {n, 7, 40}] (* Vincenzo Librandi, Jul 09 2015 *)

a(n) = binomial(2*n,7) + binomial(n,2)*binomial(2*n-4,3) - n*binomial(2*n-2,5) - (2*n-6)*binomial(n,3).
a(n) = C(n,n-7)*2^7, n>=7. - Zerinvary Lajos, Dec 07 2007
G.f.: 128*x^7/(1-x)^8. - Colin Barker, Mar 18 2012
a(n) = 128*A000580(n). a(n+1) = 2*(n+1)*a(n)/(n-6) for n >= 7. - Robert Israel, Jul 08 2015

cp's OEIS Frontend

A178820 Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.

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A130812 If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 6-subsets of X containing none of X_i, (i=1,...n).

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A172242 Number of 10-D hypercubes in an n-dimensional hypercube.

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Sage

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A130811 If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n) is equal to the number of 5-subsets of X containing none of X_i, (i=1,...n).

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Magma

Maple

Mathematica

Formula

A213345 3-quantum transitions in systems of N>=3 spin 1/2 particles, in columns by combination indices.

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Mathematica

PARI

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A248826 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x+k)^k for 0 <= k <= n.

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PARI

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A317495 Triangle read by rows: T(0,0) = 1; T(n,k) =2 * T(n-1,k) + T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

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