A122803 -id:A122803 - cp's OEIS frontend

A253145 Triangular numbers (A000217) omitting the term 1.

0, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275

Offset: 0

Paul Curtz, Mar 23 2015

The full triangle of the inverse Akiyama-Tanigawa transform applied to (-1)^n*A062510(n)=3*(-1)^n*A001045(n) yielding a(n) is
0,      3,    6,  10,   15,  21,  28, 36, ...
-3,    -6,  -12, -20,  -30, -42, -56, ...    essentially -A002378
3,     12,   24,  40,   60,  84, ...         essentially  A046092
-9,   -24,  -48, -80, -120, ...              essentially -A033996
15,    48,   96, 160, ...
-33,  -96, -192, ...
63,   192, ...
-129, ...
etc.
First column: (-1)^n*A062510(n).
The following columns are multiples of A122803(n)=(-2)^n. See A007283(n), A091629(n), A020714(n+1), A110286, A175805(n), 4*A005010(n).
An autosequence of the first kind is a sequence whose main diagonal is A000004 = 0's.
b(n) = 0, 0 followed by a(n) is an autosequence of the first kind.
The successive differences of b(n) are
0,   0,  0, 3, 6, 10, 15, 21, ...
0,   0,  3, 3, 4,  5,  6,  7, ...  see A194880(n)
0,   3,  0, 1, 1,  1,  1,  1, ...
3,  -3,  1, 0, 0,  0,  0,  0, ...
-6,  4, -1, 0, 0,  0,  0,  0, ...
10, -5,  1, 0, 0,  0,  0,  0, ...
-15, 6, -1, 0, 0,  0,  0,  0, ...
21, -7,  1, 0, 0,  0,  0,  0, ...
The inverse binomial transform (first column) is the signed sequence. This is general.
Also generalized hexagonal numbers without 1. - Omar E. Pol, Mar 23 2015

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A179865, A001045, A062510, A002378, A046092, A033996, A122803, A007283, A091629, A020714, A110286, A161680, A175805, A005010, A194880, A001840, A022003, A255935.

Concatenation([0],List([1..50],n->(n+1)*(n+2)/2)); # Muniru A Asiru, Oct 31 2018

Prepend[Table[(n + 1) (n + 2)/2, {n, 49}], 0] (* Michael De Vlieger, Mar 23 2015 *)

a(n)=if(n,(n+1)*(n+2)/2,0) \\ Charles R Greathouse IV, Mar 23 2015

Inverse Akiyama-Tanigawa transform of (-1)^n*A062510(n).
a(n) = (n+1)*(n+2)/2 for n > 0. - Charles R Greathouse IV, Mar 23 2015
a(n+1) = 3*A001840(n+1) + A022003(n).
a(n) = A161680(n+2) for n >= 1. - Georg Fischer, Oct 30 2018
From Stefano Spezia, May 28 2025: (Start)
G.f.: x*(3 - 3*x + x^2)/(1 - x)^3.
E.g.f.: exp(x)*(2 + 4*x + x^2)/2 - 1. (End)

A347237 Möbius transform of A347236.

Original entry on oeis.org

1, 0, 1, 6, 1, 0, 3, 6, 17, 0, 1, 6, 3, 0, 1, 42, 1, 0, 3, 6, 3, 0, 5, 6, 37, 0, 49, 18, 1, 0, 5, 78, 1, 0, 3, 102, 3, 0, 3, 6, 1, 0, 3, 6, 17, 0, 5, 42, 89, 0, 1, 18, 5, 0, 1, 18, 3, 0, 1, 6, 5, 0, 51, 330, 3, 0, 3, 6, 5, 0, 1, 102, 5, 0, 37, 18, 3, 0, 3, 42, 353, 0, 5, 18, 1, 0, 1, 6, 7, 0, 9, 30, 5, 0, 3, 78, 3, 0, 17

Offset: 1

Antti Karttunen, Aug 24 2021

Dirichlet convolution of A003972 (prime shifted phi) with A061019.
Dirichlet convolution of A003961 with A158523.
Multiplicative because A003972 and A061019 are, and also because A347236 is.
From Antti Karttunen, Aug 25 2021: (Start)
All terms are nonnegative because sequence is multiplicative and a(p^k) >= 0 for all primes p and k >= 0.
Proof: For any prime p, sequence a(p^k), k>=0, is obtained as an ordinary convolution of sequences (-p)^k and the first differences of q^k, where q = A151800(p). (E.g., for powers of 2, the sequences convolved are A122803 and A025192, giving A102901.) This convolution is an alternating sum, with the terms 1*(q-1)*q^(k-1), -(p)*(q-1)*q^(k-2), (p^2)*(q-1)*q^(k-3), -(p^3)*(q-1)*q^(k-4), ..., (p^(k-1))*(q-1), -(p^k), for odd k, with sum of each consecutive pair being nonnegative because q >= p+1, while with an even exponent k, the leftover term p^k at the end is also positive, thus the whole sum is nonnegative also in that case.
(End)

Cf. A000040, A001223, A003961, A003972, A008683, A008836, A016825 (positions of zeros), A061019, A102901, A120612, A158523, A347236.
Cf. also A347137.
Cf. A151800.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A061019(n) = (((-1)^bigomega(n))*n);
A347237(n) = sumdiv(n,d,A061019(d)*eulerphi(A003961(n/d)));
\\ Or alternatively as:
A158523(n) = { my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); ((-1)^e)*(p+1)*(p^(e-1))); };
A347237(n) = sumdiv(n,d,A003961(n/d)*A158523(d));

a(n) = Sum_{d|n} A008683(n/d) * A347236(d).
a(n) = Sum_{d|n} A003972(n/d) * A061019(d).
a(n) = Sum_{d|n} A003961(n/d) * A158523(d).
For all n >= 1, a(A000040(n)) = A001223(n) - 1.
For all n >= 0, a(2^n) = A102901(n).
For all n >= 0, a(3^n) = A120612(n).
Multiplicative with a(p^e) = (-p)^e + (A151800(p)-1)*(A151800(p)^e-(-p)^e)/(A151800(p)+p). - Sebastian Karlsson, Sep 02 2021

A100551 Coefficient list of ChebyshevU(n, 1-x).

Original entry on oeis.org

1, 2, -2, 3, -8, 4, 4, -20, 24, -8, 5, -40, 84, -64, 16, 6, -70, 224, -288, 160, -32, 7, -112, 504, -960, 880, -384, 64, 8, -168, 1008, -2640, 3520, -2496, 896, -128, 9, -240, 1848, -6336, 11440, -11648, 6720, -2048, 256, 10, -330, 3168, -13728, 32032, -43680, 35840, -17408, 4608, -512

Offset: 0

Wouter Meeussen, Nov 27 2004

			Triangle begins as:
  1;
  2,   -2;
  3,   -8,    4;
  4,  -20,   24,    -8;
  5,  -40,   84,   -64,    16;
  6,  -70,  224,  -288,   160,    -32;
  7, -112,  504,  -960,   880,   -384,   64;
  8, -168, 1008, -2640,  3520,  -2496,  896,  -128;
  9, -240, 1848, -6336, 11440, -11648, 6720, -2048, 256;

Cf. A001353, A001787, A053117, A056594, A122803.

[Binomial(n+k+1, n-k)*(-2)^k: k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 27 2023

Table[CoefficientList[ChebyshevU[n, 1-x], x], {n, 0, 12}]

row(n) = Vecrev(polchebyshev(n, 2, 1-x)); \\ Michel Marcus, Apr 27 2020

def A100551(n,k): return binomial(n+k+1, n-k)*(-2)^k
flatten([[A100551(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 27 2023

G.f.: ChebyshevU(n, 1-x).
From G. C. Greubel, Mar 27 2023: (Start)
T(n, k) = binomial(n+k+1, n-k)*(-2)^k.
T(n, n) = A122803(n).
T(n, n-1) = 2*(-1)^(n-1)*A001787(n), n >= 1.
Sum_{k=0..n} T(n, k) = A056594(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A001353(n+1). (End)

Keyword tabl from Michel Marcus, Apr 27 2020

A130668 Diagonal of A129819.

Original entry on oeis.org

0, 0, 1, -2, 5, -11, 23, -48, 102, -220, 476, -1024, 2184, -4624, 9744, -20480, 42976, -90048, 188352, -393216, 819328, -1704192, 3539200, -7340032, 15203840, -31456256, 65010688, -134217728, 276826112, -570429440, 1174409216

Offset: 0

Paul Curtz, Jun 27 2007

sign

This sequence is connected to A124072. To see this, change the sign of every negative term and consider the differences of every line. Hence for the second line, and following lines, the four terms form periodic sequences:
 1   0   1   0
 0   0   1   1
 0   1   2   1
 1   3   3   1
 4   6   4   2
10  10   6   6
20  16  12  16
36  28  28  36
64  56  64  72
120 120 136 136
240 256 272 256.
The lines are connected as seen by the examples: (3rd line connected to 2nd, from right to left) 1+1=2, 1+0=1, 0+0=0, 0+1=1; (11th line connected to 10th) 136+136=272, 136+120=256, 120+120=240, 120+136=256.
The 4 columns are almost known (must the first line be suppressed?): A038503 (without the first 1), A000749 (without the first 0), A038505, A038504. Like the present sequence, every sequence of A124072 beginning with a negative number (-2, -11, ...) is a "twisted" sequence (see A129339 comments, A129961 and the present 4 columns). Periodic with period 2^n.
Inverse binomial transform of A129819. - R. J. Mathar, Feb 25 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-6,-14,-16,-8).

a:=[-2,5,-11,23];; for n in [5..30] do a[n]:=-6*a[n-1]+-14*a[n-2] -16*a[n-3]-8*a[n-4]; od; Concatenation([0,0,1], a); # G. C. Greubel, Mar 24 2019

I:=[-2,5,-11,23]; [0,0,1] cat [n le 4 select I[n] else -6*Self(n-1) - 14*Self(n-2)-16*Self(n-3)-8*Self(n-4): n in [1..30]]; // G. C. Greubel, Mar 24 2019

gf = x^2*(1+x)*(1+3*x+4*x^2+3*x^3)/((1+2*x+2*x^2)*(1+2*x)^2); CoefficientList[Series[gf, {x, 0, 30}], x] (* Jean-François Alcover, Dec 16 2014, after R. J. Mathar *)
Join[{0, 0, 1}, LinearRecurrence[{-6,-14,-16,-8}, {-2,5,-11,23}, 30]] (* Jean-François Alcover, Feb 15 2016 *)

my(x='x+O('x^30)); concat([0,0], Vec(x^2*(1+x)*(1+3*x+4*x^2+3*x^3 )/((1+2*x +2*x^2)*(1+2*x)^2))) \\ G. C. Greubel, Mar 24 2019

(x^2*(1+x)*(1+3*x+4*x^2+3*x^3)/((1+2*x+2*x^2)*(1+2*x)^2 )).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Mar 24 2019

From R. J. Mathar, Feb 25 2009: (Start)
G.f.: x^2*(1+x)*(1 + 3*x + 4*x^2 + 3*x^3)/((1 + 2*x + 2*x^2)*(1+2*x)^2).
a(n) = ((-1)^n*A001787(n+1) - 4*A108520(n) + 4*A122803(n))/32, n > 2. (End)
a(n) = -6*a(n-1) - 14*a(n-2) - 16*a(n-3) - 8*a(n-4) for n >= 7. - G. C. Greubel, Mar 24 2019

Extended by R. J. Mathar, Feb 25 2009

A176260 Periodic sequence: Repeat 5, 1.

Original entry on oeis.org

5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5

Offset: 0

Klaus Brockhaus, Apr 13 2010

Interleaving of A010716 and A000012.
Also continued fraction expansion of (5+3*sqrt(5))/2.
Also decimal expansion of 17/33.
Essentially first differences of A047264.
Binomial transform of 5 followed by -A122803 without initial terms 1, -2.
Inverse binomial transform of 5 followed by A007283 without initial term 3.
Second inverse binomial transform of A168607 without initial term 3.
Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 3*x^2 + 3*x^3 + 6*x^4 + 6*x^5 + ... is the o.g.f. for A008805. - Peter Bala, Mar 13 2015

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010716 (all 5's sequence), A000012 (all 1's sequence), A090550 (decimal expansion of (5+3*sqrt(5))/2), A010686 (repeat 1, 5), A047264 (congruent to 0 or 5 mod 6), A122803 (powers of -2), A007283 (3*2^n), A168607 (3^n+2), A008805.

&cat[ [5, 1]: n in [0..52] ];
[ 3+2*(-1)^n: n in [0..104] ];

a(n) = 3+2*(-1)^n.
a(n) = a(n-2) for n > 1; a(0) = 5, a(1) = 1.
a(n) = -a(n-1)+6 for n > 0; a(0) = 5.
a(n) = 5*((n+1) mod 2)+(n mod 2).
a(n) = A010686(n+1).
G.f.: (5+x)/(1-x^2).
From Amiram Eldar, Jan 01 2023: (Start)
Multiplicative with a(2^e) = 5, and a(p^e) = 1 for p >= 3.
Dirichlet g.f.: zeta(s)*(1+2^(2-s)). (End)
E.g.f.: 5*cosh(x) + sinh(x). - Stefano Spezia, Feb 09 2025

A176414 Expansion of (7+8x)/(1+2x).

Original entry on oeis.org

7, -6, 12, -24, 48, -96, 192, -384, 768, -1536, 3072, -6144, 12288, -24576, 49152, -98304, 196608, -393216, 786432, -1572864, 3145728, -6291456, 12582912, -25165824, 50331648, -100663296, 201326592, -402653184, 805306368

Offset: 0

Klaus Brockhaus, Apr 17 2010

Inverse binomial transform of A176415.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2).

Cf. A176415, A110164 (essentially the same), A122803.

Join[{7},NestList[-2#&,-6,40]] (* Harvey P. Dale, Jun 20 2020 *)

{for(n=0, 29, print1(polcoeff((7+8*x)/(1+2*x)+x*O(x^n), n), ", "))}

A176414(n)=3*(-2)^n+!n*4 \\ M. F. Hasler, Apr 19 2015

a(n) = A110164(n+2) for n > 0.
a(n) = 3*(-2)^n = 3*A122803(n+1) for n > 0; a(0) = 7.
a(n) = -2*a(n-1) for n > 1; a(0) = 7, a(1) = -6.
a(n) = (-1)^n*A132477(n) = (-1)^n*A122391(n+3), n>1.
a(n) = (-1)^n*A111286(n+2) = (-1)^n*A098011(n+4) = (-1)^n*A091629(n) = (-1)^n*A087009(n+3) = (-1)^n*A082505(n+1) = (-1)^n*A042950(n+1) = (-1)^n*A007283(n) = (-1)^n*A003945(n+1), n>0. - R. J. Mathar, Dec 10 2010
E.g.f.: 4 + 3*exp(-2*x). - Alejandro J. Becerra Jr., Feb 15 2021

Edited by M. F. Hasler, Apr 19 2015

A316995 Sequence of distinct signed integers such that a(1) = 0 and for any n > 0, a(n+1) is of the form a(n) + (-2)^k (where k >= 0) and has the smallest possible absolute value (in case of a tie, minimize k).

Original entry on oeis.org

0, 1, -1, -3, -2, 2, 3, 4, -4, -6, -5, -7, -9, 7, 5, 6, 10, 8, 9, 13, 11, 12, 16, 14, 15, -17, -13, -12, -8, -10, -18, -14, -16, -15, -11, -19, -21, -20, -22, -24, -23, -25, -27, -26, -28, -30, -29, -31, -33, 31, 23, 21, 19, 17, 18, 22, 20, 24, 25, 26, 27, 28

Offset: 1

Rémy Sigrist, Jul 18 2018

sign

This sequence is likely to contain every signed integer.

			The first terms, alongside the value k such that a(n+1) = a(n) + (-2)^k, are:
  n  a(n)   k
  -- ----   --
   1    0    0
   2    1    1
   3   -1    1
   4   -3    0
   5   -2    2
   6    2    0
   7    3    0
   8    4    3
   9   -4    1
  10   -6    0
  11   -5    1
  12   -7    1
  13   -9    4
  14    7    1
  15    5    0
  16    6    2
  17   10    1
  18    8    0
  19    9    2
  20   13    1

Rémy Sigrist, PARI program for A316995

Cf. A122803.

PARI
```
See Links section.
```

A318223 Expansion of e.g.f. exp(x/(1 + 2*x)).

Original entry on oeis.org

1, 1, -3, 13, -71, 441, -2699, 9157, 206193, -8443151, 236126701, -6169406979, 161388751657, -4327824442967, 120012465557349, -3450029411174219, 102741264191105761, -3160671409312412703, 99982488984008583133, -3230094912866216253971, 105481073534842477321881, -3423260541695907002392679

Offset: 0

Ilya Gutkovskiy, Aug 21 2018

sign

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A002866, A025168, A111884, A122803.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x/(1+2*x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Feb 07 2019

seq(n!*coeff(series(exp(x/(1+2*x)),x=0,22),x,n),n=0..21); # Paolo P. Lava, Jan 09 2019

nmax = 21; CoefficientList[Series[Exp[x/(1 + 2 x)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-2)^(n - k) Binomial[n - 1, k - 1] n!/k!, {k, 0, n}], {n, 0, 21}]
a[n_] := a[n] = Sum[(-2)^(k - 1) k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]
Join[{1}, Table[(-2)^(n - 1) n! Hypergeometric1F1[1 - n, 2, 1/2], {n, 21}]]

my(x='x+O('x^30)); Vec(serlaplace(exp(x/(1+2*x)))) \\ G. C. Greubel, Feb 07 2019

m = 30; T = taylor(exp(x/(1+2*x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Feb 07 2019

E.g.f.: Product_{k>=1} exp((-2)^(k-1)*x^k).
a(n) = Sum_{k=0..n} (-2)^(n-k)*binomial(n-1,k-1)*n!/k!.
a(0) = 1; a(n) = Sum_{k=1..n} (-2)^(k-1)*k!*binomial(n-1,k-1)*a(n-k).

A335111 a(n) = n! * Sum_{k=0..n-1} (-2)^k / k!.

Original entry on oeis.org

0, 1, -2, 6, -8, 40, 48, 784, 5248, 49536, 490240, 5403904, 64822272, 842742784, 11798284288, 176974510080, 2831591636992, 48137058942976, 866467058614272, 16462874118651904, 329257482362552320, 6914407129635618816, 152116956851937476608, 3498690007594658430976

Offset: 0

Ilya Gutkovskiy, May 23 2020

sign

Inverse binomial transform of A000240.

Cf. A000023, A000240, A008290, A066534, A087981, A122803.

Table[n! Sum[(-2)^k/k!, {k, 0, n - 1}], {n, 0, 23}]
nmax = 23; CoefficientList[Series[Sum[k! x^k/(1 + 2 x)^(k + 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 23; CoefficientList[Series[x Exp[-2 x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

a(n) = n! * sum(k=0, n-1, (-2)^k / k!); \\ Michel Marcus, May 23 2020

G.f.: Sum_{k>=1} k! * x^k / (1 + 2*x)^(k + 1).
E.g.f.: x*exp(-2*x) / (1 - x).
a(n) = A000023(n) - A122803(n).
a(n) ~ exp(-2) * n!. - Vaclav Kotesovec, Jun 08 2022
a(n) = Sum_{k=0..n} (-1)^k * k * A008290(n,k). - Alois P. Heinz, May 20 2023

A381082 Triangle T(n,k) read by rows, where the columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)exp(-mx)/k! for the case m=2.

Original entry on oeis.org

1, -2, 1, 4, -3, 1, -8, 8, -3, 1, 16, -18, 11, -2, 1, -32, 44, -20, 15, 0, 1, 64, -80, 94, 5, 25, 3, 1, -128, 272, 56, 294, 105, 49, 7, 1, 256, 112, 1868, 1596, 1169, 392, 98, 12, 1, -512, 5280, 12216, 16148, 10290, 4305, 1092, 186, 18, 1

Offset: 0

Igor Victorovich Statsenko, Feb 13 2025

			Triangle starts:
  [0]     1;
  [1]    -2,      1;
  [2]     4,     -3,       1;
  [3]    -8,      8,      -3,       1;
  [4]    16,    -18,      11,      -2,       1;
  [5]   -32,     44,     -20,      15,       0,        1;
  [6]    64,    -80,      94,       5,      25,        3,     1;
  [7]  -128,    272,      56,     294,     105,       49,     7,     1;
  [8]   256,    112,    1868,    1596,    1169,      392,    98,    12,    1;
  [9]  -512,   5280,   12216,   16148,   10290,     4305,  1092,   186,   18,     1;
  ...

Cf. A000023 (row sums).
Columns 0,1: A122803, A346397.
Triangles: for m = -3 is A327997; for m = -2 is A137346 (unsigned); for m = -1 is A094816; for m = 0 is A132393; for m = 1 is A269953.

T:=(m,n,k)->add(Stirling1(n-i,k)*binomial(n,i)*m^(i)*(-1)^(n-k), i=0..n):
m:=2:seq(print(seq(T(m,n,k), k=0..n)), n=0..9);

T(n,k) = Sum_{i=0..n} Stirling1(n-i, k)*binomial(n, i)*m^(i)*(-1)^(n-k), where m = 2.

cp's OEIS Frontend

A253145 Triangular numbers (A000217) omitting the term 1.

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A347237 Möbius transform of A347236.

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A100551 Coefficient list of ChebyshevU(n, 1-x).

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A130668 Diagonal of A129819.

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A176260 Periodic sequence: Repeat 5, 1.

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A176414 Expansion of (7+8*x)/(1+2*x).

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A316995 Sequence of distinct signed integers such that a(1) = 0 and for any n > 0, a(n+1) is of the form a(n) + (-2)^k (where k >= 0) and has the smallest possible absolute value (in case of a tie, minimize k).

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A176414 Expansion of (7+8x)/(1+2x).

A381082 Triangle T(n,k) read by rows, where the columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)exp(-mx)/k! for the case m=2.