A011853 -id:A011853 - cp's OEIS frontend

A011857 Triangle of numbers [ C(n,k)/k ], k=1..n-1.

2, 3, 1, 4, 3, 1, 5, 5, 3, 1, 6, 7, 6, 3, 1, 7, 10, 11, 8, 4, 1, 8, 14, 18, 17, 11, 4, 1, 9, 18, 28, 31, 25, 14, 5, 1, 10, 22, 40, 52, 50, 35, 17, 5, 1, 11, 27, 55, 82, 92, 77, 47, 20, 6, 1, 12, 33, 73, 123, 158, 154, 113, 61, 24, 6, 1, 13, 39, 95, 178, 257, 286, 245, 160

Offset: 2

N. J. A. Sloane

Columns include A011848, A011849, A011850, A011851, A011852, A011853, A011854, A011855, A011856. Row sums are in A101687. Cf. A011847.

Flatten[Table[Floor[Binomial[n,k]/k],{n,20},{k,n-1}]] (* Harvey P. Dale, Apr 19 2015 *)

A011797 a(n) = floor(C(n,6)/7).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 4, 12, 30, 66, 132, 245, 429, 715, 1144, 1768, 2652, 3876, 5537, 7752, 10659, 14421, 19228, 25300, 32890, 42287, 53820, 67860, 84825, 105183, 129456, 158224, 192129, 231880, 278256

Offset: 0

N. J. A. Sloane, David Broadhurst

a(n-1) is the number of aperiodic necklaces (Lyndon words) with 7 black beads and n-7 white beads.

David J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
Index entries for sequences related to Lyndon words
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1,1,-6,15,-20,15,-6,1).

Cf. A000031, A001037, A051168. Same as A051172(n+1).
First differences of A011853.
A column of triangle A011847.

CoefficientList[Series[x^6/7 (1/(1-x)^7-1/(1- x^7)),{x,0,40}],x]; (* Herbert Kociemba, Oct 16 2016 *)

a(n) = binomial(n, 6)\7; \\ Michel Marcus, Oct 16 2016

G.f.: (1+x^3)^2/((1-x)^4(1-x^2)^2(1-x^7))*x^7.
a(n) = floor(binomial(n+1,7)/(n+1)). [Gary Detlefs, Nov 23 2011]
G.f.: (x^6/7)*(1/(1-x)^7-1/(1- x^7)). - Herbert Kociemba, Oct 16 2016

A215053 a(n) = 1/7*( binomial(n,7) - floor(n/7) ).

Original entry on oeis.org

1, 5, 17, 47, 113, 245, 490, 919, 1634, 2778, 4546, 7198, 11074, 16611, 24363, 35022, 49443, 68671, 93971, 126861, 169148, 222968, 290828, 375653, 480836, 610292, 768516, 960645, 1192525, 1470781, 1802893, 2197276, 2663365, 3211705, 3854046

Offset: 8

Peter Bala, Aug 01 2012

Not the same as A011853.

Let p be a prime. Saikia and Vogrinc have proved that 1/p*{binomial(n,p) - floor(n/p)} is an integer sequence. The present sequence is the case p = 7. Other cases are A002620 (p = 2), A014125 (p = 3), A215052 (p = 5) and A215054 (p = 11).

Alexandre Laugier and Manjil Saikia, A characterization of a prime p from the binomial coefficient binomial(n,p), arXiv:1209.2373 [math.NT], The Mathematics Student, Vol. 83 (1-4), pp 1-7, 2014
M. P. Saikia and J. Vogrinc, A simple number theoretic result, arXiv:1207.6707v1 [math.NT]
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 2, -7, 21, -35, 35, -21, 7, -1).

Cf. A002620 (p = 2), A014125 (p = 3), A178904, A215052 (p = 5), A215054 (p = 11).

[(Binomial(n, 7)-Floor(n/7))/7: n in [8..50]]; // Vincenzo Librandi, Jun 23 2015

Table[(Binomial[n,7]-Floor[n/7])/7,{n,8,50}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,2,-7,21,-35,35,-21,7,-1},{1,5,17,47,113,245,490,919,1634,2778,4546,7198,11074,16611},40] (* Harvey P. Dale, Dec 23 2014 *)

a(n) = (binomial(n, 7) - n\7) / 7; \\ Michel Marcus, Jan 23 2014

O.g.f.: sum {n>=0} a(n)*x^n = x^8*(1 - 2*x + 3*x^2 - 2*x^3 + x^4)/((1-x^7)*(1-x)^7) = x^8*(1 + 5*x + 17*x^2 + 47*x^3 + ...). The numerator polynomial 1 - 2*x + 3*x^2 - 2*x^3 + x^4 is the negative of the row generating polynomial for row 7 of A178904.

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A011857 Triangle of numbers [ C(n,k)/k ], k=1..n-1.

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A011797 a(n) = floor(C(n,6)/7).

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A215053 a(n) = 1/7*( binomial(n,7) - floor(n/7) ).

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