A020063 -id:A020063 - cp's OEIS frontend

A020060 a(n) = floor( Gamma(n+9/10)/Gamma(9/10) ).

1, 0, 1, 4, 19, 94, 559, 3857, 30477, 271252, 2685395, 29270806, 348322596, 4493361500, 62457724853, 930620100318, 14796859595058, 250066927156481, 4476197996101023, 84600142126309335, 1683542828313555774

Offset: 0

Simon Plouffe

nonn

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A020015, A020105.

Cf. A020061, A020062, A020063.

[Floor(Gamma(n+9/10)/Gamma(9/10)): n in [0..20]]; // G. C. Greubel, Nov 13 2019

Digits := 64:f := proc(n,x) trunc(GAMMA(n+x)/GAMMA(x)); end;
seq(floor(pochhammer(9/10,n)), n = 0..20); # G. C. Greubel, Nov 13 2019

Floor[Pochhammer[9/10, Range[0, 20]]] (* G. C. Greubel, Nov 13 2019 *)

vector(21, n, my(x=9/10); gamma(n-1+x)\gamma(x) ) \\ G. C. Greubel, Nov 13 2019

[floor(rising_factorial(9/10, n)) for n in (0..20)] # G. C. Greubel, Nov 13 2019

A020018 Nearest integer to Gamma(n + 1/10)/Gamma(1/10).

Original entry on oeis.org

1, 0, 0, 0, 1, 3, 15, 91, 649, 5253, 47802, 482796, 5359036, 64844333, 849460756, 11977396665, 180858689642, 2911824903230, 49792205845229, 901238925798638, 17213663482753993, 345994636003355265, 7300486819670796088

Offset: 0

Simon Plouffe

nonn

			Gamma(1/10)/Gamma(1/10) = 1, so a(0) = 1.
Gamma(1 + 1/10)/Gamma(1/10) = 1/10 < 1/2, so a(1) = 0.
Gamma(2 + 1/10)/Gamma(1/10) = 11/100 < 1/2, so a(2) = 0.
Gamma(3 + 1/10)/Gamma(1/10) = 231/1000 < 1/2, so a(3) = 0.
Gamma(4 + 1/10)/Gamma(1/10) = 7161/10000 = 0.7161, so a(4) = 1.
Gamma(5 + 1/10)/Gamma(1/10) = 293601/100000 = 2.93601, so a(5) = 3.
Gamma(6 + 1/10)/Gamma(1/10) = 14973651/1000000 = 14.973651, so a(6) = 15.

G. C. Greubel, Table of n, a(n) for n = 0..450

Cf. A045757, A020063, A020108, A000007 (decimal expansion of 1/10), A256191 (decimal expansion of Gamma(1/10)).

[Round(Gamma(n +1/10)/Gamma(1/10)): n in [0..30]]; // G. C. Greubel, Jan 20 2018

Digits := 64:f := proc(n,x) round(GAMMA(n+x)/GAMMA(x)); end;

Table[Round[Gamma[n + 1/10]/Gamma[1/10]], {n, 0, 50}] (* G. C. Greubel, Jan 20 2018 *)

for(n=0,30, print1(round(gamma(n+1/10)/gamma(1/10)), ", ")) \\ G. C. Greubel, Jan 20 2018

A072148 Number of invertible (-1,0,1) n X n matrices having (Tij = -Tji; i

Original entry on oeis.org

2, 14, 92, 796, 7672, 83944

Offset: 1

Wouter Meeussen, Aug 25 2003

The matrix powers T^k reach identity I for k a divisor of 12. All T^k are invertible (-1,0,1)-matrices with determinant +/-1. The matrix |Tij| is symmetric. The matrices T are "pseudo-anti-symmetric" (that is Tij=-Tji except for the main diagonal, or, equivalently, the sum of an anti-symmetric and a diagonal matrix). Their eigenvalues belong to {-1, -I, I, 1, -(-1)^(1/3), (-1)^(1/3), -(-1)^(2/3), (-1)^(2/3)}.

			{{1,-1,0,0,0},{1,0,0,0,0},{0,0,0,-1,0},{0,0,1,1,0},{0,0,0,0,-1}}
qualifies since its powers are:
{{0,-1,0,0,0},{1,-1,0,0,0},{0,0,-1,-1,0},{0,0,1,0,0},{0,0,0,0,1}},
{{-1,0,0,0,0},{0,-1,0,0,0},{0,0,-1,0,0},{0,0,0,-1,0},{0,0,0,0,-1}},
{{-1,1,0,0,0},{-1,0,0,0,0},{0,0,0,1,0},{0,0,-1,-1,0},{0,0,0,0,1}},
{{0,1,0,0,0},{-1,1,0,0,0},{0,0,1,1,0},{0,0,-1,0,0},{0,0,0,0,-1}},
{{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,0},{0,0,0,1,0},{0,0,0,0,1}}.

Cf. A081959 A002464 A020063 A033169 A090410 A066052.

triamatsig[li_List] := Block[{len=Sqrt[8Length[li]+1]/2-1/2}, If[IntegerQ[len], (Part[li, # ]&/@ Table[If[j>i, j(j-1)/2+i, i(i-1)/2+j], {i, len}, {j, len}])Table[If[j>i, -1, 1], {i, len}, {j, len}], li]]; n=4; it=triamatsig/@(-1+IntegerDigits[Range[0, -1+3^(n(n+1)/2)], 3, n(n+1)/2]); result4=Cases[it, (q_?MatrixQ)/; Det[q]=!=0 && And@@ Table[Union[Flatten[{MatrixPower[q, k], {-1, 0, 1}}]]==={-1, 0, 1}, {k, 25}]]

a(6) from Wouter Meeussen, Nov 15 2005

A182831 Joint-rank array of numbers j*r^(i-1), where r=1+sqrt(2), read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 11, 17, 22, 7, 14, 28, 45, 55, 9, 19, 37, 70, 112, 137, 10, 23, 48, 93, 171, 276, 334, 12, 26, 57, 118, 228, 417, 671, 812, 13, 31, 66, 141, 287, 556, 1010, 1627, 1965, 15, 34, 77, 164, 344, 697, 1347, 2444, 3934, 4751, 16, 39

Offset: 1

Clark Kimberling, Dec 07 2010

Joint-rank arrays are defined in the first comment at A182801. (row 1)=A087063. First 3 columns are A020062, A020063, A020064.

Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.

			Northwest corner:
   1  2  4  5 ...
   3  6 11 14 ...
   8 17 28 37 ...
  22 45 70 93 ...
  ...

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A182801.

T[n_, k_] := Sum[Floor[k*(1 + Sqrt[2])^(n - j)], {j, 1, 100}]; Table[T[k + 1, n - k], {n,1,10}, {k, 0, n-1}]//Flatten (* G. C. Greubel, Aug 18 2018 *)

T(i,j) = Sum_{n>=1} floor(j*(1+sqrt(2))^(i-n)).

cp's OEIS Frontend

A020060 a(n) = floor( Gamma(n+9/10)/Gamma(9/10) ).

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A020018 Nearest integer to Gamma(n + 1/10)/Gamma(1/10).

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A072148 Number of invertible (-1,0,1) n X n matrices having (Tij = -Tji; i

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A182831 Joint-rank array of numbers j*r^(i-1), where r=1+sqrt(2), read by antidiagonals.

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