A247917 -id:A247917 - cp's OEIS frontend

A047916 Triangular array read by rows: a(n,k) = phi(n/k)(n/k)^kk! if k|n else 0 (1<=k<=n).

1, 2, 2, 6, 0, 6, 8, 8, 0, 24, 20, 0, 0, 0, 120, 12, 36, 48, 0, 0, 720, 42, 0, 0, 0, 0, 0, 5040, 32, 64, 0, 384, 0, 0, 0, 40320, 54, 0, 324, 0, 0, 0, 0, 0, 362880, 40, 200, 0, 0, 3840, 0, 0, 0, 0, 3628800, 110, 0, 0, 0, 0, 0, 0, 0, 0, 0, 39916800, 48, 144

Offset: 1

N. J. A. Sloane

T(n,k) = A054523(n,k) * A010766(n,k)^A002260(n,k) * A166350(n,k). - Reinhard Zumkeller, Jan 20 2014

			1; 2,2; 6,0,6; 8,8,0,24; 20,0,0,0,120; 12,36,48,0,0,720; ...

J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.
N. J. A. Sloane, Notes on A002618, A002619, etc.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61. [Annotated scanned copy. Note that the scanned pages are out of order]

A064649 gives the row sums.
Cf. A002618 (left edge), A000142 (right edge), A049820 (zeros per row), A000005 (nonzeros per row).
See also A247917, A047918, A047919.

import Data.List (zipWith4)
a047916 n k = a047916_tabl !! (n-1) !! (k-1)
a047916_row n = a047916_tabl !! (n-1)
a047916_tabl = zipWith4 (zipWith4 (\x u v w -> x * v ^ u * w))
               a054523_tabl a002260_tabl a010766_tabl a166350_tabl
-- Reinhard Zumkeller, Jan 20 2014

a[n_, k_] := If[Divisible[n, k], EulerPhi[n/k]*(n/k)^k*k!, 0]; Flatten[ Table[ a[n, k], {n, 1, 12}, {k, 1, n}]] (* Jean-François Alcover, May 04 2012 *)

a(n,k)=if(n%k, 0, eulerphi(n/k)*(n/k)^k*k!) \\ Charles R Greathouse IV, Feb 09 2017

A104769 Expansion of g.f. -x/(1+x-x^3).

Original entry on oeis.org

0, -1, 1, -1, 0, 1, -2, 2, -1, -1, 3, -4, 3, 0, -4, 7, -7, 3, 4, -11, 14, -10, -1, 15, -25, 24, -9, -16, 40, -49, 33, 7, -56, 89, -82, 26, 63, -145, 171, -108, -37, 208, -316, 279, -71, -245, 524, -595, 350, 174, -769, 1119, -945, 176, 943, -1888, 2064, -1121, -767, 2831, -3952

Offset: 0

Creighton Dement, Mar 24 2005

Generating floretion is "jesright".

Pisano period lengths: 1, 7, 13, 14, 24, 91, 48, 28, 39, 168, 120, 182, 183, 336, 312, 56, 288, 273, 180, 168,.. (which differs from A104217 for example at index 23). - R. J. Mathar, Aug 10 2012

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (-1,0,1).

Apart from signs, essentially the same as A050935 and A078013.
Cf. A247917 (negative).
Cf. A000931, A057597.

LinearRecurrence[{-1, 0, 1}, {0, -1, 1}, 61] (* or *)
CoefficientList[Series[-x/(1 + x - x^3), {x, 0, 60}], x] (* Michael De Vlieger, Jul 02 2021 *)

a(n)=([0,1,0;0,0,1;1,0,-1]^n*[0;-1;1])[1,1] \\ Charles R Greathouse IV, Jun 11 2015

a(n) = -A247917(n-1).
Recurrence: a(n+3) = a(n) - a(n+2); a(0) = 0, a(1) = -1, a(2) = 1.
a(n+1) - a(n) = ((-1)^(n+1))*a(n+5).
a(n) = ((-1)^n)*A050935(n+1) = ((-1)^n)*A078013(n+2).
a(n) = A104771(n) - A104770(n).

Edited by Ralf Stephan, Apr 05 2009

A247919 Expansion of 1 / (1 + x^4 - x^5) in powers of x.

Original entry on oeis.org

1, 0, 0, 0, -1, 1, 0, 0, 1, -2, 1, 0, -1, 3, -3, 1, 1, -4, 6, -4, 0, 5, -10, 10, -4, -5, 15, -20, 14, 1, -20, 35, -34, 13, 21, -55, 69, -47, -8, 76, -124, 116, -39, -84, 200, -240, 155, 45, -284, 440, -395, 110, 329, -724, 835, -505, -219, 1053, -1559, 1340

Offset: 0

Michael Somos, Sep 26 2014

			G.f. = 1 - x^4 + x^5 + x^8 - 2*x^9 + x^10 - x^12 + 3*x^13 - 3*x^14 + x^15 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,-1,1).

Cf. A003520, A010892, A247917.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1 + x^4 - x^5)));  // G. C. Greubel, Aug 04 2018

CoefficientList[Series[1/(1 + x^4 - x^5), {x, 0, 100}], x] (* Vincenzo Librandi, Sep 27 2014 *)
LinearRecurrence[{0,0,0,-1,1},{1,0,0,0,-1},60] (* Harvey P. Dale, Sep 11 2024 *)

{a(n) = if( n<0, n=-5-n; polcoeff( 1 / (1 - x - x^5) + x * O(x^n), n), polcoeff( 1 / (1 + x^4 - x^5) + x * O(x^n), n))};

G.f.: 1 / ((1 - x + x^2) * (1 + x - x^3)).
Convolution of A010892 and A247917.
a(-5-n) = A003520(n) for all n in Z.
0 = a(n) - a(n+1) - a(n+5) for all n in Z.

A257543 Expansion of 1 / (1 - x^5 - x^8 + x^9) in powers of x.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 1, -1, 1, 0, 0, 2, -2, 1, 1, -2, 4, -3, 1, 3, -6, 7, -3, -2, 9, -13, 11, -1, -11, 22, -23, 12, 10, -33, 46, -35, 2, 43, -78, 81, -37, -41, 122, -159, 118, 4, -162, 281, -277, 114, 167, -443, 558, -391, -52, 610, -1001, 949, -338, -662

Offset: 0

Michael Somos, Apr 28 2015

			G.f. = 1 + x^5 + x^8 - x^9 + x^10 + 2*x^13 - 2*x^14 + x^15 + x^16 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1,0,0,1,-1).

Cf. A104769, A233522, A247917, A247918.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1 / ((1-x^4)*(1+x^4-x^5)))); // G. C. Greubel, Aug 02 2018

a[ n_] := SeriesCoefficient[ If[ n >= 0, 1 / (1 - x^5 - x^8 + x^9), -x^9 /(1 - x - x^4 + x^9)], {x, 0, Abs@n}];

{a(n) = if( n>=0, polcoeff( 1 / (1 - x^5 - x^8 + x^9) + x * O(x^n), n), polcoeff( -x^9 / (1 - x - x^4 + x^9) + x * O(x^-n), -n))};

G.f.: 1 / ((1 - x^4) * (1 + x^4 - x^5)) = (1 + x) / ((1 + x^3) * (1 - x^4) * (1 + x - x^3)).
a(n) = a(n-5) + a(n-8) - a(n-9) for all n in Z.
a(n) - a(n+2) - a(n+3) has period 12.
a(n) - a(n+12) = A104769(n+5) = -A247917(n+4) for all n in Z.
a(n) + a(n+1) = A247918(n) for all n in Z.
a(n) = -A233522(-9 - n) for all n in Z.

A358613 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * (n-k)!/(k! * (n-3*k)!).

Original entry on oeis.org

1, 1, 1, -1, -5, -11, -7, 31, 139, 245, -71, -1937, -5989, -6251, 25945, 144479, 304843, -177899, -3517351, -11743505, -10097381, 81902453, 433558201, 840235039, -1481279605, -15839941451, -48073840007, -8454966289, 564429256219, 2518098130645, 3490609807769

Offset: 0

Seiichi Manyama, Nov 23 2022

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A247917, A358560.

a(n) = sum(k=0, n\3, (-1)^k*(n-k)!/(k!*(n-3*k)!));

a(n) = (4 * a(n-1) - a(n-2) - 2 * (2*n-3) * a(n-3))/3 for n > 2.

cp's OEIS Frontend

A047916 Triangular array read by rows: a(n,k) = phi(n/k)*(n/k)^k*k! if k|n else 0 (1<=k<=n).

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A104769 Expansion of g.f. -x/(1+x-x^3).

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A247919 Expansion of 1 / (1 + x^4 - x^5) in powers of x.

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A257543 Expansion of 1 / (1 - x^5 - x^8 + x^9) in powers of x.

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A358613 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * (n-k)!/(k! * (n-3*k)!).

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A047916 Triangular array read by rows: a(n,k) = phi(n/k)(n/k)^kk! if k|n else 0 (1<=k<=n).