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A179096 Rectified hexateron (5-simplex) numbers: the coefficient of x^(2n-2) in (1+x+x^2+...+x^(n-1))^6.

0, 1, 15, 90, 336, 951, 2247, 4676, 8856, 15597, 25927, 41118, 62712, 92547, 132783, 185928, 254864, 342873, 453663, 591394, 760704, 966735, 1215159, 1512204, 1864680, 2280005, 2766231, 3332070, 3986920, 4740891, 5604831, 6590352

Offset: 0

Michael A. Jackson, Jun 29 2010

a(n) is the number of ordered 6-tuples (j_1,...,j_6) with 0 <= j_i <= n-1 and Sum_{i=1..6} j_i = 2n-2. - Robert Israel, Feb 17 2016

J. Conrad and Robert Israel, Table of n, a(n) for n = 0..1000 (n = 0..98 from J. Conrad)
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A179095, A179097, A179098, A179099.

F:= n -> coeff(add(x^i,i=0..n-1)^6,x,2*n-2):
seq(F(n),n=0..100); # Robert Israel, Feb 17 2016

f[n_] := CoefficientList[ Series[ Sum[x^k, {k, 0, n - 1}]^6, {x, 0, 2 n + 3}], x][[2 n - 1]]; Array[f, 36] (* Robert G. Wilson v, Jul 30 2010 *)

a(n) = polcoeff(((x^n-1)/(x-1))^6, 2*n-2); \\ Michel Marcus, Feb 17 2016

Conjectures: a(n) = n*(n+1)*(13*n^3+12*n^2-7*n+12)/60. G.f.: x*(1+9*x+x^3+15*x^2)/(x-1)^6. - R. J. Mathar, Jul 06 2010

These conjectures are true, see A179095 for proof.

A179097 Rectified heptapeton (6-simplex) numbers: the coefficient of x^(2n-2) in (1+x+x^2+...+x^(n-1))^7.

Original entry on oeis.org

0, 1, 21, 161, 728, 2415, 6538, 15330, 32292, 62601, 113575, 195195, 320684, 507143, 776244, 1154980, 1676472, 2380833, 3316089, 4539157, 6116880, 8127119, 10659902, 13818630, 17721340, 22502025, 28312011, 35321391, 43720516

Offset: 0

Michael A. Jackson, Jun 29 2010

J. Conrad, Table of n, a(n) for n = 0..53
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A179095, A179096, A179098, A179099.

f[n_] := CoefficientList[ Series[ Sum[x^k, {k, 0, n - 1}]^7, {x, 0, 2 n + 3}], x][[2 n - 1]]; Array[f, 33] (* Robert G. Wilson v, Jul 30 2010 *)

a(n) = polcoeff(((x^n-1)/(x-1))^7, 2*n-2); \\ Michel Marcus, Feb 17 2016

Conjecture: a(n) = n*(40+26*n+5*n^2+75*n^3+75*n^4+19*n^5)/240. G.f.: x*(1+14*x+35*x^2+7*x^3)/(1-x)^7. - Colin Barker, Jan 09 2012

These conjectures are true, see A179095 for proof.

More terms from Robert G. Wilson v, Jul 30 2010

A179098 Rectified 7-simplex number: the coefficient of x^(2n-2) in (1+x+x^2+...+x^(n-1))^8.

Original entry on oeis.org

0, 1, 28, 266, 1428, 5475, 16808, 44052, 102552, 217701, 429220, 796510, 1405196, 2374983, 3868944, 6104360, 9365232, 14016585, 20520684, 29455282, 41534020, 57629099, 78796344, 106302780, 141656840, 186641325, 243349236, 314222598

Offset: 0

Michael A. Jackson, Jun 29 2010

nonn

J. Conrad, Table of n, a(n) for n = 0..34
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A179095, A179096, A179097, A179099.

f[n_] := CoefficientList[ Series[ Sum[ x^k, {k, 0, n - 1}]^8, {x, 0, 2 n + 3}], x][[2 n - 1]]; Array[f, 33] (* Robert G. Wilson v, Jul 30 2010 *)

a(n) = polcoeff(((x^n-1)/(x-1))^8, 2*n-2); \\ Michel Marcus, Feb 17 2016

Conjectures: a(n) = n*(90+77*n+140*n^3+210*n^4+98*n^5+15*n^6)/630. G.f.: x*(1+20*x+70*x^2+28*x^3+x^4)/(1-x)^8. - Colin Barker, Jan 09 2012

These conjectures are true, see A179095 for proof.

More terms from Robert G. Wilson v, Jul 30 2010

A179099 Rectified 8-simplex numbers: the coefficient of x^(2n-2) in (1+x+x^2+...+x^(n-1))^9.

Original entry on oeis.org

0, 1, 36, 414, 2598, 11385, 39303, 114387, 292743, 677556, 1446445, 2889315, 5459103, 9838062, 17022474, 28428930, 46025562, 72491859, 111410946, 167498452, 246872340, 357368319, 508905705, 713908845, 987789465, 1349495550

Offset: 0

Michael A. Jackson, Jun 29 2010

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A179095, A179096, A179097, A179098.

f[n_] := CoefficientList[ Series[ Sum[ x^k, {k, 0, n - 1}]^9, {x, 0, 2 n + 3}], x][[2 n - 1]]; Array[f, 33] (* Robert G. Wilson v, Jul 30 2010 *)

a(n) = polcoeff(((x^n-1)/(x-1))^9, 2*n-2); \\ Michel Marcus, Feb 17 2016

Empirical G.f.: x*(1+27*x+126*x^2+84*x^3+9*x^4)/(1-x)^9. - Colin Barker, Jun 20 2012

This conjecture is true, see A179095 for proof.

More terms from Robert G. Wilson v, Jul 30 2010

A179095 Rectified 5-cell numbers: the coefficient of x^{2n-2} in (1+x+x^2+ ... + x^{n-1})^5.

Original entry on oeis.org

0, 1, 10, 45, 135, 320, 651, 1190, 2010, 3195, 4840, 7051, 9945, 13650, 18305, 24060, 31076, 39525, 49590, 61465, 75355, 91476, 110055, 131330, 155550, 182975, 213876, 248535, 287245, 330310, 378045, 430776, 488840, 552585, 622370, 698565, 781551, 871720, 969475

Offset: 0

Michael A. Jackson, Jun 29 2010

J. Conrad, Table of n, a(n) for n = 0..260
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A179096, A179097, A179098, A179099.

f[n_] := CoefficientList[ Series[ Sum[x^k, {k, 0, n - 1}]^5, {x, 0, 2 n + 3}], x][[2 n - 1]]; Array[f, 36] (* Robert G. Wilson v, Jul 30 2010 *)

a(n) = polcoeff(((x^n-1)/(x-1))^5, 2*n-2); \\ Michel Marcus, Feb 17 2016

A179095(n)=n*(11*n^3+6*n^2+n+6)\24 \\ M. F. Hasler, Feb 19 2016

Conjectures: a(n) = n*(11*n^3+6*n^2+n+6)/24. G.f.: x*(1+5*x+5*x^2)/(1-x)^5. - Colin Barker, Jan 09 2012
Comment from Doron Zeilberger, Feb 18 2016 (Start):
The conjectures in A179095-A179099 are true. Proof:
The geometric series 1+x+x^2+..+x^(n-1) = (1-x^n)/(1-x).
Hence for a fixed k (in the above cases k=5..9, but the argument holds in general)
the coefficient of  x^(2*n-2) in (1+x+...+x^(n-1))^k =
coefficient of  x^(2*n-2) in (1-x^n)^k*(1-x)^(-k) =
coefficient of  x^(2*n-2) in (1-k*x^n + Sum of powers higher than x^(2*n-2)..)
= coefficient of x^(2*n-2) in (1-x)^(-k) -k*(the coefficient of x^(n-2) in (1-x)^(-k))
= (-1)^(2*n-2)*binomial(-k,2*n-2)- k* (-1)^(n-2)*binomial(-k,n-2)=
Using  (-1)^m *binomial(-m,k)= binomial(m+k-1,k-1) this is
binomial(k+2*n-3,k-1) - k *binomial(k+n-3,k-1)
and this agrees with the conjectures for k=5..9 (End)
E.g.f.: exp(x)*x*(24 + 96*x + 72*x^2 + 11*x^3)/24. - Stefano Spezia, Mar 28 2023

More terms from Robert G. Wilson v, Jul 30 2010

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User: Michael A. Jackson

A179096 Rectified hexateron (5-simplex) numbers: the coefficient of x^(2n-2) in (1+x+x^2+...+x^(n-1))^6.

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A179097 Rectified heptapeton (6-simplex) numbers: the coefficient of x^(2n-2) in (1+x+x^2+...+x^(n-1))^7.

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A179098 Rectified 7-simplex number: the coefficient of x^(2n-2) in (1+x+x^2+...+x^(n-1))^8.

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A179099 Rectified 8-simplex numbers: the coefficient of x^(2n-2) in (1+x+x^2+...+x^(n-1))^9.

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A179095 Rectified 5-cell numbers: the coefficient of x^{2n-2} in (1+x+x^2+ ... + x^{n-1})^5.

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