cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Milan Janjic, Jul 16 2007

Keywords

Comments

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

Links

Crossrefs

Cf. A038207, A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354, A046092, A130809, A130810, A130811, A130812. - Zerinvary Lajos, Aug 05 2008
Cf. A000580.

Programs

  • Magma
    [Binomial(n,n-7)*2^7: n in [7..40]]; // Vincenzo Librandi, Jul 09 2015
  • Maple
    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
  • Mathematica
    Table[Binomial[n, n - 7] 2^7, {n, 7, 40}] (* Vincenzo Librandi, Jul 09 2015 *)

Formula

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

A185342 Triangle of successive recurrences in columns of A117317(n).

Original entry on oeis.org

2, 4, -4, 6, -12, 8, 8, -24, 32, -16, 10, -40, 80, -80, 32, 12, -60, 160, -240, 192, -64, 14, -84, 280, -560, 672, -448, 128, 16, -112, 448, -1120, 1792, -1792, 1024, -256, 18, -144, 672, -2016, 4032, -5376, 4608, -2304, 512, 20, -180, 960, -3360, 8064
Offset: 0

Views

Author

Paul Curtz, Jan 26 2012

Keywords

Comments

A117317 (A):
1
2 1
4 5 1
8 16 9 1
16 44 41 14 1
32 112 146 85 20 1
64 272 456 377 155 27 1
have for their columns successive signatures
(2) (4,-4) (6,-12,8) (8,-24, 32, -16) (10,-40,80,-80,32) i.e. a(n).
Take based on abs(A133156) (B):
1
2 0
4 1 0
8 4 0 0
16 12 1 0 0
32 32 6 0 0 0
64 80 24 1 0 0 0.
The recurrences of successive columns are also a(n). a(n) columns: A005843(n+1), A046092(n+1), A130809, A130810, A130811, A130812, A130813.
A053220 + A001787 = A014480.

Examples

			Triangle T(n,k),for 1<=k<=n, begins :
2                                         (1)
4    -4                                   (2)
6   -12   8                               (3)
8   -24  32   -16                         (4)
10  -40  80   -80   32                    (5)
12  -60 160  -240  192   -64              (6)
14  -84 280  -560  672  -448  128         (7)
16 -112 448 -1120 1792 -1792 1024 -256    (8)
Successive rows can be divided by A171977.

Links

Crossrefs

Cf. For (A): A053220, A056243. For (B): A000079, A001787, A001788, A001789. For A193862: A115068 (a Coxeter group). For (2): A014480 (also (3),(4),(5),..); also A053220 and A001787.
Cf. A007318.

Programs

  • Mathematica
    Table[(-1)*Binomial[n, k]*(-2)^k, {n, 1, 20}, {k, 1, n}] // Flatten (* G. C. Greubel, Jun 27 2017 *)
  • PARI
    for(n=1,20, for(k=1,n, print1((-2)^(k+1)*binomial(n,k)/2, ", "))) \\ G. C. Greubel, Jun 27 2017

Formula

Take A133156(n) without 1's or -1's double triangle (C)=
2
4
8 -4
16 -12
32 -32 6
64 -80 24
128 -192 80 -8
256 -448 240 -40
512 -1024 672 -160 10;
a(n) is increasing odd diagonals and increasing (sign changed) even diagonals. Rows sum of (C) = A201629 (?) Another link between Chebyshev polynomials and cos( ).
Absolute values: A013609(n) without 1's. Also 2*A193862 = (2*A002260)*A135278.
T(n,k) = T(n-1,k) - 2*T(n-1,k-1) for k>1, T(n,1) = 2*n = 2*T(n-1,1) - T(n-2,1). - Philippe Deléham, Feb 11 2012
T(n,k) = (-1)* Binomial(n,k)*(-2)^k, 1<=k<=n. - Philippe Deléham, Feb 11 2012

A373547 Triangle read by rows: T(n,k) = 4^k*binomial(n+k, n-k) with 0 <= k <= n.

Original entry on oeis.org

1, 1, 4, 1, 12, 16, 1, 24, 80, 64, 1, 40, 240, 448, 256, 1, 60, 560, 1792, 2304, 1024, 1, 84, 1120, 5376, 11520, 11264, 4096, 1, 112, 2016, 13440, 42240, 67584, 53248, 16384, 1, 144, 3360, 29568, 126720, 292864, 372736, 245760, 65536, 1, 180, 5280, 59136, 329472, 1025024, 1863680, 1966080, 1114112, 262144
Offset: 0

Views

Author

Stefano Spezia, Jun 09 2024

Keywords

Comments

T(n,k) is the number of occurrences of the periodic substring (01)^k in the periodic string (0011)^n (see Proposition 4.3 at page 6 in Fang).
The word (w_1, w_2, ..., w_r)^m is defined as the word obtained by concatenating (w_1, w_2, ..., w_r) m times.
A word w' = (w'1, w'_2, ..., w'_s) is said be a subword of a given word w = (w_1, w_2, ..., w_r), if there is some set P = {p_1 < ... < p_s} of integers from 1 to r satisfying w{p_j} = w'_j for all 1 <= j <= s, and we call the set P an occurrence of w' in w (see Preliminaries section at pp. 2-3 in Fang).

Examples

			The triangle begins as:
  1;
  1,  4;
  1, 12,   16;
  1, 24,   80,   64;
  1, 40,  240,  448,   256;
  1, 60,  560, 1792,  2304,  1024;
  1, 84, 1120, 5376, 11520, 11264, 4096;
  ...
T(2,1) = 12 since there are 12 occurrences of (01)^1 = 01 in (0011)^2 = 00110011: {1, 3}, {1, 4}, {1, 7}, {1, 8}, {2, 3}, {2, 4}, {2, 7}, {2, 8}, {5, 7}, {5, 8}, {6, 7}, {6, 8}.

Links

Crossrefs

Cf. A000012 (k=0), A000302 (diagonal), A001653 (row sums), A046092 (k=1), A046717, A085478, A130810, A130812, A373628.

Programs

  • Mathematica
    T[n_,k_]:=4^k Binomial[n+k,n-k]; Table[T[n,k],{n,0,9},{k,0,n}]//Flatten (* or *)
T[n_,k_]:=SeriesCoefficient[(1-x)/((1-x)^2-4x y),{x,0,n},{y,0,k}]; Table[T[n,k],{n,0,9},{k,0,n}]//Flatten

Formula

G.f.: (1 - x)/((1 - x)^2 - 4*x*y).
T(n,k) = A000302(k)*A085478(n,k).
Sum_{k=0..n} T(n-k,k) = A046717(n).
T(n,2) = A130810(n+2).
T(n,3) = A130812(n+3).
Showing 1-3 of 3 results.

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