A127276 -id:A127276 - cp's OEIS frontend

A176126 Numerator of -A127276(n)/A001788(n).

-1, -1, 1, 2, 4, 13, 19, 13, 17, 43, 53, 32, 38, 89, 103, 59, 67, 151, 169, 94, 104, 229, 251, 137, 149, 323, 349, 188, 202, 433, 463, 247, 263, 559, 593, 314, 332, 701, 739, 389, 409, 859, 901, 472, 494, 1033, 1079, 563, 587, 1223, 1273, 662, 688, 1429, 1483, 769, 797, 1651, 1709, 884, 914, 1889, 1951, 1007, 1039, 2143, 2209, 1138, 1172, 2413, 2483, 1277, 1313, 2699, 2773, 1424, 1462, 3001, 3079, 1579

Offset: 0

Paul Curtz, Dec 07 2010

The sequence of fractions starts -1/0, -1/1, 1/3, 2/3, 4/5, 13/15, 19/21, 13/14, 17/18, 43/45, 53/55, 32/33, 38/39, ...
The denominators are apparently A064038(n+1) = A061041(4+8*n) (i.e., specified as numerators in A061041).
The difference between denominator and numerator is A014695(n), n > 0.

Cf. A001788, A127276.

Cf. A061041, A064038, A014695.

A001788 := proc(n) n*(n+1)*2^(n-2) ; end proc:
A127276 := proc(n) 2^n-A001788(n) ; end proc:
A176126 := proc(n) if n = 0 then -1 else 2^n/A001788(n)-1 ; numer(-%) ; end if; end proc:
seq(A176126(n),n=0..40) ;

Conjecture: a(n) = +3*a(n-1) -6*a(n-2) +10*a(n-3) -12*a(n-4) +12*a(n-5) -10*a(n-6) +6*a(n-7) -3*a(n-8) +a(n-9) with g.f. (x^4-x^3+3*x^2-x+1)*(x^4-x^3-2*x^2-x+1) / ( (x-1)^3*(x^2+1)^3 ). - R. J. Mathar, Dec 12 2010

a(n) = 3*a(n-4) -3*a(n-8) +a(n-12).

A178987 a(n) = n(n-3)2^(n-2).

Original entry on oeis.org

0, -1, -2, 0, 16, 80, 288, 896, 2560, 6912, 17920, 45056, 110592, 266240, 630784, 1474560, 3407872, 7798784, 17694720, 39845888, 89128960, 198180864, 438304768, 964689920, 2113929216, 4613734400, 10032775168, 21743271936, 46976204800, 101200166912

Offset: 0

Paul Curtz, Jan 03 2011

Binomial transform of 0, -1 followed by A005563.
The sequence defines an array by adding higher order differences in successive rows:
0, -1, -2, 0, 16, 80, 288, 896, 2560, 6912, 17920, 45056, 110592, ...
-1, -1, 2, 16, 64, 208, 608, 1664, 4352, 11008, 27136, 65536, ... A127276
0, 3, 14, 48, 144, 400, 1056, 2688, 6656, 16128, 38400, 90112, 208896, ... A176027
3, 11, 34, 96, 256, 656, 1632, 3968, 9472, 22272, 51712, 118784, ... A084266
8, 23, 62, 160, 400, 976, 2336, 5504, 12800, 29440, 67072, ...
The left column of the array (binomial transform of the sequence) is A067998.
For n>2, the sequence gives the number of permutations in the symmetric group S_{n+1} with peaks exactly in positions 2 and n-1. See Theorem 10 in [Billey-Burdzy-Sagan] reference.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Sara Billey, Krzystof Burdzy and Bruce Sagan, Permutations With Given Peak Set, J. Integer Sequences, Vol. 16 (2013), online article 13.6.1.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A176027.

[n*(n-3)*2^(n-2): n in [0..30]]; // Vincenzo Librandi, Aug 04 2011

Table[n(n-3)2^(n-2),{n,0,30}] (* or *) LinearRecurrence[{6,-12,8},{0,-1,-2},30] (* Harvey P. Dale, Mar 24 2023 *)

a(n) = 16*A001793(n-3), n > 3.
a(n) = 8*A001788(n-2)-A052481(n-1). - R. J. Mathar, Jan 04 2011
a(n) = +6*a(n-1) -12*a(n-2) +8*a(n-3).
a(n+1)-a(n) = -A127276(n).
G.f.: -x*(-1+4*x)/(2*x-1)^3. - R. J. Mathar, Jan 04 2011
a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} (k-1) * C(n-1,i). - Wesley Ivan Hurt, Sep 20 2017
a(n) = Sum_{k=0..n} k^2 * (-1)^k * 3^(n-k) * binomial(n,k). - Seiichi Manyama, Apr 18 2025

A127275 Expansion of (sqrt(1-4x)-x)/(1-4x).

Original entry on oeis.org

1, 1, 2, 4, 6, -4, -100, -664, -3514, -16916, -77388, -343144, -1490148, -6376616, -26992264, -113317936, -472661434, -1961361076, -8104733884, -33374212936, -137031378124, -561253753336, -2293947547384, -9358755316816, -38121140494564, -155064370272904

Offset: 0

Paul D. Hanna, Jun 07 2006

Hankel transform is A127276.

The second self-composition of the g.f. G(x) of A120009 is G(G(x)) = (sqrt(1-4x)-x)/(1-4x) - 1.

			A(x) = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 - 4*x^5 - 100*x^6 - 664*x^7 + ...

Robert Israel, Table of n, a(n) for n = 0..1653

Cf. A120009, A120012 (3rd self-composition); A000108 (Catalan).

S:= series((sqrt(1-4*x)-x)/(1-4*x),x,31):
seq(coeff(S,x,i),i=0..30); # Robert Israel, Jan 15 2023

{a(n)=local(k=2,x=X+X^3*O(X^n));polcoeff( x*((1-k+k^2)-k^2*(k+1)*x-k*(1-(k+2)*x)*(1-sqrt(1-4*x))/2/x)/(1-k+k^2*x)^2,n,X)}

a(n) = C(2n,n) - 4^(n-1) + 0^n/4. - Paul Barry, Jan 10 2007
Conjecture: n*a(n) + 2*(-4*n+3)*a(n-1) + 8*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 26 2012
Conjecture verified using the differential equation (4*x-1)^2 * g'(x) + (8*x-2)*g(x) + 1 - 2*x = 0 satisfied by the g.f. - Robert Israel, Jan 15 2023

Definition revised by Paul Barry, Jan 10 2007

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar and Max Alekseyev

A176027 Binomial transform of A005563.

Original entry on oeis.org

0, 3, 14, 48, 144, 400, 1056, 2688, 6656, 16128, 38400, 90112, 208896, 479232, 1089536, 2457600, 5505024, 12255232, 27131904, 59768832, 131072000, 286261248, 622854144, 1350565888, 2919235584, 6291456000, 13522436096

Offset: 0

Paul Curtz, Dec 06 2010

The numbers appear on the diagonal of a table T(n,k), where the left column contains the elements of A005563, and further columns are recursively T(n,k) = T(n,k-1)+T(n-1,k-1):
....0....-1.....0.....0.....0.....0.....0.....0.....0.....0.
....3.....3.....2.....2.....2.....2.....2.....2.....2.....2.
....8....11....14....16....18....20....22....24....26....28.
...15....23....34....48....64....82...102...124...148...174.
...24....39....62....96...144...208...290...392...516...664.
...35....59....98...160...256...400...608...898..1290..1806.
...48....83...142...240...400...656..1056..1664..2562..3852.
...63...111...194...336...576...976..1632..2688..4352..6914.
...80...143...254...448...784..1360..2336..3968..6656.11008.
...99...179...322...576..1024..1808..3168..5504..9472.16128.
..120...219...398...720..1296..2320..4128..7296.12800.22272.
The second column is A142463, the third A060626, the fourth essentially A035008 and the fifth essentially A016802. Transposing the array gives A005563 and its higher order differences in the individual rows.

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A001792, A001793, A005563, A058396, A084266, A127276.

[2^(n-2)*n*(5+n) : n in [0..30]]; // Vincenzo Librandi, Oct 08 2011

LinearRecurrence[{6,-12,8},{0,3,14},30] (* Harvey P. Dale, Oct 19 2015 *)

a(n)=n*(n+5)<<(n-2) \\ Charles R Greathouse IV, Sep 21 2017

G.f.: x*(-3+4*x)/(2*x-1)^3. - R. J. Mathar, Dec 11 2010
a(n) = 2^(n-2)*n*(5+n). - R. J. Mathar, Dec 11 2010
a(n) = A127276(n) - A127276(n+1).
a(n+1)-a(n) = A084266(n+1).
a(n+2) = 16*A058396(n) for n > 0.
a(n) = 2*a(n-1) + A001792(n).
a(n) = A001793(n) - 2^(n-1) for n > 0. - Brad Clardy, Mar 02 2012
a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} (k+3) * C(n-1,i). - Wesley Ivan Hurt, Sep 20 2017
From Amiram Eldar, Aug 13 2022: (Start)
Sum_{n>=1} 1/a(n) = 1322/75 - 124*log(2)/5.
Sum_{n>=1} (-1)^(n+1)/a(n) = 132*log(3/2)/5 - 782/75. (End)

A181407 a(n) = (n-4)(n-3)2^(n-2).

Original entry on oeis.org

3, 3, 2, 0, 0, 16, 96, 384, 1280, 3840, 10752, 28672, 73728, 184320, 450560, 1081344, 2555904, 5963776, 13762560, 31457280, 71303168, 160432128, 358612992, 796917760, 1761607680, 3875536896, 8489271296, 18522046464, 40265318400, 87241523200, 188441690112

Offset: 0

Paul Curtz, Jan 28 2011

Binomial transform of (3, 0, -1, followed by A005563).
The sequence and its successive differences are:
   3,  3,  2,   0,   0,   16,   96,  384,      a(n),
   0, -1, -2,   0,  16,   80,  288,  896,      A178987,
  -1, -1,  2,  16,  64,  208,  608, 2688,     -A127276,
   0,  3, 14,  48, 144,  400, 1056, 2688,      A176027,
   3, 11, 34,  96, 256,  656, 1632, 3968,      A084266(n+1)
   8, 23, 62, 160, 400,  976, 2336, 5504,
  15, 39, 98, 240, 576, 1360, 3168, 7296.
Division of the k-th column by abs(A174882(k)) gives
   3,  3,  1,  0,  0,  1,  3,  3,  5,  15,  21, 14,   A064038(n-3),
   0, -1, -1,  0,  1,  5,  9,  7, 10,  27,  35, 22,   A160050(n-3),
  -1, -1,  1,  2,  4, 13, 19, 13, 17,  43,  53, 32,   A176126,
   0,  3,  7,  6,  9, 25, 33, 21, 26,  63,  75, 44,   A178242,
   3, 11, 17, 12, 16, 41, 51, 31, 37,  87, 101, 58,
   8  23, 31, 20, 25, 61, 73, 43, 50, 115, 131, 74,
  15, 39, 49, 30, 36, 85, 99, 57, 65, 147, 165, 92.
Columns are (or from) A005563, A142463, A056220, A002378, A000290, A001844, A058331, A002061, A002522, A097080, A093328, A014206.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A176027, A181318.

List([0..40], n-> (n-4)*(n-3)*2^(n-2)); # G. C. Greubel, Feb 21 2019

[(n-4)*(n-3)*2^(n-2): n in [0..40] ]; // Vincenzo Librandi, Feb 01 2011

Table[(n-4)*(n-3)*2^(n-2), {n,0,40}] (* G. C. Greubel, Feb 21 2019 *)

vector(40, n, n--; (n-4)*(n-3)*2^(n-2)) \\ G. C. Greubel, Feb 21 2019

[(n-4)*(n-3)*2^(n-2) for n in (0..40)] # G. C. Greubel, Feb 21 2019

a(n) = 16*A001788(n-4).
a(n+1) - a(n) = A178987(n).
G.f.: (3 - 15*x + 20*x^2) / (1-2*x)^3. - R. J. Mathar, Jan 30 2011
E.g.f.: (x^2 - 3*x + 3)*exp(2*x). - G. C. Greubel, Feb 21 2019

A236999 Odd part of n*(n+3)/2-1 (A034856).

Original entry on oeis.org

1, 1, 1, 13, 19, 13, 17, 43, 53, 1, 19, 89, 103, 59, 67, 151, 169, 47, 13, 229, 251, 137, 149, 323, 349, 47, 101, 433, 463, 247, 263, 559, 593, 157, 83, 701, 739, 389, 409, 859, 901, 59, 247, 1033, 1079, 563, 587, 1223, 1273, 331, 43, 1429, 1483

Offset: 1

Vladimir Shevelev, Feb 02 2014

Also odd part of A176126(n-1) and of |A127276(n-1)|, n>=3.
Proof. By A127276 and A001788, we have odd part(A176126(n))=odd part(|A127276(n)|) = odd part(n*(n+1)-4), {odd part(A176126(n-1)), n>=3}={odd part((n+1)*(n+2)-4), n>=1}.
Let n=2^b*k, where k=k(n) is odd.
Then {odd part(A176126(n-1)), n>=3}={odd part((2^b*k+1)*(2^b*k+2)-4)}={odd part(2^(2*b)*k^2+3*2^b*k-2)}. Hence, if b>0, then {odd part(A176126(n-1), n>=3)= {odd part(2^(2*b-1)*k^2+3*2^(b-1)*k-1)}.
On the other hand, in this case odd part(a(n))=odd part(2^(b-1)*k*(2^b*k+3)-1)=odd part(2^(2*b-1)*k^2+3*2^(b-1)*k-1). It is left to consider the case of odd n. Setting n=2*m-1, m>=1, we easily find that for both expressions the odd part equals odd part(2*m^2+m-2).
The smallest prime divisor of a(n) is more than or equal to 13.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000265, A034856, A176126, A127276, A001788.

Map[#/2^IntegerExponent[#,2]&[(# (#+3)/2-1)]&,Range[100]] (* Peter J. C. Moses, Feb 02 2014 *)

a(n) = A000265(A034856(n)). - Michel Marcus, Feb 25 2025

cp's OEIS Frontend

A176126 Numerator of -A127276(n)/A001788(n).

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A178987 a(n) = n*(n-3)*2^(n-2).

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A127275 Expansion of (sqrt(1-4x)-x)/(1-4x).

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A176027 Binomial transform of A005563.

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A181407 a(n) = (n-4)*(n-3)*2^(n-2).

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A236999 Odd part of n*(n+3)/2-1 (A034856).

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A178987 a(n) = n(n-3)2^(n-2).

A181407 a(n) = (n-4)(n-3)2^(n-2).