A375302 -id:A375302 - cp's OEIS frontend

A138523 a(n) = Sum_{k=1..n} (2k-1)!.

1, 7, 127, 5167, 368047, 40284847, 6267305647, 1313941673647, 357001369769647, 122002101778601647, 51212944273488041647, 25903229683158464681647, 15537113273014144448681647, 10904406563691366305216681647, 8852666400303393320848832681647, 8231691320578226211046411712681647

Offset: 1

Leroy Quet, Mar 23 2008

a(n) is divisible by 107 for n >= 53. - Robert Israel, Dec 01 2015
Last digit is 7 for n > 1. Therefore there is no square in this sequence except 1. - Altug Alkan, Dec 01 2015
a(n) is the rank of [2,1, 4,3, ..., 2n,2n-1] within the permutations of [1, 2, ... 2n-1, 2n] in lexicographic order. See A375302 for the ranking function. - Hugo Pfoertner, Aug 25 2024

G. C. Greubel, Table of n, a(n) for n = 1..225

Cf. A007489, A138524, A138525, A375302.

a:=proc(n) options operator, arrow: sum(factorial(2*k-1), k=1..n) end proc: seq(a(n), n=1..14); # Emeric Deutsch, Mar 31 2008

Table[Sum[(2i - 1)!, {i, n}], {n, 15}] (* Stefan Steinerberger, Mar 25 2008 *)

a(n) = sum(k=1, n, (2*k-1)!); \\ Michel Marcus, Oct 28 2015

Recurrence: a(1) = 1, a(2) = 7, a(n) = (4*n^2-6*n+3)*a(n-1) - 2*(n-1)*(2*n-1)*a(n-2). - Vladimir Reshetnikov, Oct 28 2015

More terms from Stefan Steinerberger, Emeric Deutsch and Robert G. Wilson v, Mar 25 2008

A375301 Triangle T(n,k) read by rows with row n equal to the unique permutation pi_n of [1, ..., n] such that k + pi_n(k) is a power of 2 for 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 3, 2, 1, 3, 2, 1, 4, 1, 2, 5, 4, 3, 1, 6, 5, 4, 3, 2, 7, 6, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 1, 8, 1, 6, 5, 4, 3, 2, 9, 8, 7, 1, 2, 5, 4, 3, 10, 9, 8, 7, 6, 3, 2, 1, 4, 11, 10, 9, 8, 7, 6, 5, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 1, 2, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3

Offset: 1

Hugo Pfoertner, Aug 25 2024

			The triangle begins
  1,
  1, 2,
  3, 2, 1,
  3, 2, 1, 4,
  1, 2, 5, 4, 3,
  1, 6, 5, 4, 3, 2,
  7, 6, 5, 4, 3, 2, 1,
  7, 6, 5, 4, 3, 2, 1, 8,
  ...
Row 5: [1, 2, 5, 4, 3] + [1, 2, 3, 4, 5] = [2, 4, 8, 8, 8]; only powers of 2 in the vector of sums.

Zhi-Wei Sun, On permutations of {1, ..., n} and related topics, J Algebr Comb 54, 893-912 (2021); Theorem 1.2, page 895.

A375302 gives the rank of row n in lexicographically ordered permutations of [n].

Cf. A000079.

a375301_row(n) = forperm(n, p, my(f=1); for(k=1, n, my(s=p[k]+k); if(2^valuation(s,2)!=s, f=0; break)); if(f==1, return(Vec(p))))

A375303 a(n) is the rank of row n of A130517 in a lexicographic permutation of [1, ..., n].

Original entry on oeis.org

0, 1, 4, 20, 108, 678, 4848, 39264, 355920, 3575640, 39454560, 474501600, 6178566240, 86606881200, 1300352981760, 20821540239360, 354184575816960, 6378546460970880, 121243261343500800, 2425719783585369600, 50955334461183398400, 1121303792572973856000, 25795667534014525593600

Offset: 1

Hugo Pfoertner, Aug 26 2024

nonn

Cf. A130517, A375302.

\\ uses functions a130517_row from A130517 and rank from A375302.
a(n) = rank(a130517_row(n))

cp's OEIS Frontend

A138523 a(n) = Sum_{k=1..n} (2k-1)!.

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A375301 Triangle T(n,k) read by rows with row n equal to the unique permutation pi_n of [1, ..., n] such that k + pi_n(k) is a power of 2 for 1 <= k <= n.

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A375303 a(n) is the rank of row n of A130517 in a lexicographic permutation of [1, ..., n].

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