A095263 -id:A095263 - cp's OEIS frontend

A136302 Transform of A000027 by the T_{1,1} transformation (see link).

2, 6, 15, 35, 81, 188, 437, 1016, 2362, 5491, 12765, 29675, 68986, 160373, 372822, 866706, 2014847, 4683951, 10888865, 25313540, 58846841, 136802308, 318026782, 739322571, 1718716457, 3995531011, 9288482690, 21593102505, 50197873146, 116695897118, 271285047567

Offset: 1

Richard Choulet, Mar 22 2008

G. C. Greubel, Table of n, a(n) for n = 1..1000
Richard Choulet, Curtz-like transformation.
Index entries for linear recurrences with constant coefficients, signature (3,-2,1).

Cf. A095263, A136303, A136304, A136305.

I:=[2,6,15]; [n le 3 select I[n] else 3*Self(n-1) -2*Self(n-2) +Self(n-3): n in [1..41]]; // G. C. Greubel, Apr 12 2021

a:= n-> (<<6|2|1>>. <<3|1|0>, <-2|0|1>, <1|0|0>>^n)[1, 3]:
seq(a(n), n=1..40);  # Alois P. Heinz, Aug 14 2008

LinearRecurrence[{3,-2,1}, {2,6,15}, 41] (* G. C. Greubel, Apr 12 2021 *)

def A136302_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(2+x^2)/(1-3*x+2*x^2-x^3) ).list()
a=A136302_list(41); a[1:] # G. C. Greubel, Apr 12 2021

G.f.: z*(2 + z^2)/(1 - 3*z + 2*z^2 - z^3).
a(n+3) = 3*a(n+2) - 2*a(n+1) + a(n) (n>=0). - Richard Choulet, Apr 07 2009
a(n) = 2*A095263(n) + A095263(n-2). - R. J. Mathar, Feb 29 2016

More terms from Alois P. Heinz, Aug 14 2008

A136303 Expansion of g.f. (1 +x^2)/((1-x)^2(1 -3x +2*x^2 -x^3)).

Original entry on oeis.org

1, 5, 17, 48, 123, 300, 714, 1679, 3925, 9149, 21296, 49537, 115192, 267824, 622653, 1447533, 3365149, 7823068, 18186475, 42278476, 98285586, 228486323, 531166317, 1234811937, 2870589548, 6673311137, 15513566304, 36064666240, 83840177305

Offset: 0

Richard Choulet, Mar 22 2008

Previous name: Transform of 0 by the reciprocal transformation to T_{1,1} (see link).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Richard Choulet, Curtz-like transformation.
Index entries for linear recurrences with constant coefficients, signature (5,-9,8,-4,1).

Cf. A095263, A097550, A135364, A136302, A136304, A136305, A137229, A137234, A137249.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x^2)/((1-x)^2*(1-3*x+2*x^2-x^3)) )); // G. C. Greubel, Apr 19 2021

A136303:= n-> -2*(n+2) + add( (5*binomial(n+k+2, 3*k+2) - 4*binomial(n +k+1, 3*k+2) + 2*binomial(n+k, 3*k+2)), k=0..n/2 );
seq(A136303(n), n=0..40); # G. C. Greubel, Apr 19 2021

LinearRecurrence[{5,-9,8,-4,1},{1,5,17,48,123},40] (* Harvey P. Dale, Apr 01 2018 *)

def A136303_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)/((1-x)^2*(1-3*x+2*x^2-x^3)) ).list()
A136303_list(40) # G. C. Greubel, Apr 19 2021

G.f.: f(z) = 1 +5*z + ... = (1+z^2)/((1-z)^2*(1-3*z+2*z^2-z^3)).
a(n+5) = 5*a(n+4) -9*a(n+3) +8*a(n+2) -4*a(n+1) +a(n) (n>=0). - Richard Choulet, Apr 07 2009
From G. C. Greubel, Apr 19 2021: (Start)
a(n) = -2*(n+2) + 5*A095263(n) - 4*A095263(n-1) + 2*A095263(n-2).
a(n) = -2*(n+2) + Sum_{k=0..floor(n/2)} (5*binomial(n+k+2, 3*k+2) - 4*binomial(n +k+1, 3*k+2) + 2*binomial(n+k, 3*k+2)). (End)

A137229 Expansion of g.f. x/((1-x)(1-3x+2*x^2-x^3)).

Original entry on oeis.org

1, 4, 11, 27, 64, 150, 350, 815, 1896, 4409, 10251, 23832, 55404, 128800, 299425, 696080, 1618191, 3761839, 8745216, 20330162, 47261894, 109870575, 255418100, 593775045, 1380359511, 3208946544, 7459895656, 17342153392, 40315615409, 93722435100, 217878227875

Offset: 1

Richard Choulet, Apr 05 2008

Previous name was: Transform of A000217 without the initial 0 by the T_{0,0} transformation (see link).

Partial sums of A095263. - R. J. Mathar, Nov 04 2008

G. C. Greubel, Table of n, a(n) for n = 1..1000
Richard Choulet, Curtz-like transformation.
Index entries for linear recurrences with constant coefficients, signature (4,-5,3,-1).

Cf. A136302, A136303, A136304, A136305.

I:=[1,4,11,27]; [n le 4 select I[n] else 4*Self(n-1) -5*Self(n-2) +3*Self(n-3) -Self(n-4): n in [1..40]]; // G. C. Greubel, Apr 17 2021

a:= n-> (<<3|1|0|0>, <-2|0|1|0>, <1|0|0|0>, <1|0|0|1>>^n)[4, 1]:
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 24 2008

LinearRecurrence[{4,-5,3,-1},{1,4,11,27},40] (* Harvey P. Dale, Nov 10 2014 *)

def A137229_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x/((1-x)*(1-3*x+2*x^2-x^3)) ).list()
a=A137229_list(41); a[1:] # G. C. Greubel, Apr 17 2021

O.g.f: x/((1-x)*(1 -3*x +2*x^2 -x^3)).

a(n) = term (4,1) in the 4x4 matrix [3,1,0,0; -2,0,1,0; 1,0,0,0; 1,0,0,1]^(n). - Alois P. Heinz, Jul 24 2008

New name using g.f., Joerg Arndt, Apr 18 2021

A368475 Expansion of o.g.f. (1-x)^5/((1-x)^5 - x^4).

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 15, 35, 71, 136, 265, 550, 1211, 2732, 6126, 13485, 29191, 62648, 134408, 289656, 627401, 1363124, 2963186, 6434484, 13951852, 30221185, 65442625, 141745045, 307137901, 665732417, 1443184210, 3128438335, 6780867186, 14696002913, 31848721632

Offset: 0

Enrique Navarrete, Dec 26 2023

For n > 0, a(n) is the number of ways to split [n] into an unspecified number of intervals and then choose 4 blocks (i.e., subintervals) from each interval. For example, for n=12, a(12)=1211 since the number of ways to split [12] into intervals and then select 4 blocks from each interval is C(12,4) + C(8,4)*C(4,4) + C(7,4)*C(5,4) + C(6,4)*C(6,4) + C(5,4)*C(7,4) + C(4,4)*C(8,4) + C(4,4)*C(4,4)*C(4,4) for a total of 1211 ways.
For n > 0, a(n) is also the number of compositions of n using parts of size at least 4 where there are binomial(i,4) types of i, i >= 4 (see example).
Number of compositions of 5*n-4 into parts 4 and 5. - Seiichi Manyama, Feb 01 2024

			Since there are C(4,4) = 1 type of 4, C(5,4) = 5 types of 5, C(6,4) = 15 types of 6, C(7,4) = 35 types of 7, C(8,4) = 70 types of 8, and (12,4) = 495 types of 12, we can write 12 in the following ways:
  12: 495 ways;
  8+4: 70 ways;
  7+5: 175 ways;
  6+6: 225 ways;
  5+7: 175 ways;
  4+8: 70 ways;
  4+4+4: 1 way, for a total of 1211 ways.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-4,1).

Cf. A000332, A017827, A099131, A290998.
Cf. A079675, A369803, A369804.
Cf. A088305, A095263, A290998, A369794, A369809.

CoefficientList[Series[(1 - x)^5/((1 - x)^5 - x^4), {x, 0, 50}], x] (* Wesley Ivan Hurt, Dec 26 2023 *)

Vec((1-x)^5/((1-x)^5 - x^4) + O(x^40)) \\ Michel Marcus, Dec 27 2023

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 4*a(n-4) + a(n-5), n>=6; a(0)=1, a(1)=a(2)=a(3)=0, a(4)=1, a(5)=5.
G.f.: 1/(1-Sum_{k>=4} binomial(k,4)*x^k).
G.f.: 1/p(S), where p(S) = 1 - S^4 - S^5 and S = x/(1-x).
First differences of A099131. - R. J. Mathar, Jan 29 2024
a(n) = A017827(5*n-4) = Sum_{k=0..floor((5*n-4)/4)} binomial(k,5*n-4-4*k) for n > 0. - Seiichi Manyama, Feb 01 2024
a(n) = Sum_{k=0..floor(n/4)} binomial(n-1+k,n-4*k). - Seiichi Manyama, Feb 02 2024

A127893 Riordan array (1/(1-x)^3, x/(1-x)^3).

Original entry on oeis.org

1, 3, 1, 6, 6, 1, 10, 21, 9, 1, 15, 56, 45, 12, 1, 21, 126, 165, 78, 15, 1, 28, 252, 495, 364, 120, 18, 1, 36, 462, 1287, 1365, 680, 171, 21, 1, 45, 792, 3003, 4368, 3060, 1140, 231, 24, 1, 55, 1287, 6435, 12376, 11628, 5985, 1771, 300, 27, 1

Offset: 0

Paul Barry, Feb 04 2007

Inverse is A127894.
From Peter Bala, Jul 22 2014: (Start)
Let M denote the unsigned version of the lower unit triangular array A122432 and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/
having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... (which is clearly well-defined). See the Example section. (End)

			Triangle begins
   1;
   3,    1;
   6,    6,     1;
  10,   21,     9,     1;
  15,   56,    45,    12,     1;
  21,  126,   165,    78,    15,     1;
  28,  252,   495,   364,   120,    18,     1;
  36,  462,  1287,  1365,   680,   171,    21,    1;
  45,  792,  3003,  4368,  3060,  1140,   231,   24,   1;
  55, 1287,  6435, 12376, 11628,  5985,  1771,  300,  27,  1;
  66, 2002, 12870, 31824, 38760, 26334, 10626, 2600, 378, 30, 1;
  ...
From _Peter Bala_, Jul 22 2014: (Start)
With the arrays M(k) as defined in the Comments section, the infinite product M(0*)M(1)*M(2)*... begins
  / 1         \/1         \/1       \       / 1       \
  | 3  1      ||0  1      ||0 1      |      | 3  1    |
  | 6  3 1    ||0  3 1    ||0 0 1    |... = | 6  6 1  |
  |10  6 3 1  ||0  6 3 1  ||0 0 3 1  |      |10 21 9 1|
  |15 10 6 3 1||0 10 6 3 1||0 0 6 3 1|      |...      |
  |...        ||...       ||...      |      |...      |
(End)

G. C. Greubel, Rows n = 0..99, flattened
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

Cf. A052529, A095263, A122432, A127894.

Flat(List([0..10],n->List([0..n],k->Binomial(n+2*k+2,n-k)))); # Muniru A Asiru, Apr 30 2018

[Binomial(n+2*k+2, n-k): k in [0..n], n in [0..10]]; // G. C. Greubel, Apr 29 2018

seq(seq(binomial(n+2*k+2,n-k),k=0..n),n=0..10); # Robert Israel, Apr 28 2015

Flatten@ Table[Binomial[n+2k-1, n-k], {n, 10}, {k, n}] (* Michael De Vlieger, Apr 27 2015 *)

for(n=0,10, for(k=0,n, print1(binomial(n+2*k+2, n-k), ", "))) \\ G. C. Greubel, Apr 29 2018

flatten([[binomial(n+2*k+2,n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 16 2021

T(n,k) = binomial(n+2*k+2, n-k).
Sum_{k=0..n} T(n, k) = A052529(n+1) (row sums).
Sum_{k=0..floor(n/2)} T(n-k, k) = A095263(n+1) (diagonal sums).
Recurrence: T(n+1, k+1) = Sum_{i = 0..n-k} binomial(i+2, 2)*T(n-i,k). - Peter Bala, Jul 22 2014
G.f.: 1/((1-x)^3-x*y). - Vladimir Kruchinin, Apr 27 2015

A137234 Expansion of g.f. 1/((1-x)^2(1 - 3x + 2*x^2 - x^3)).

Original entry on oeis.org

1, 5, 16, 43, 107, 257, 607, 1422, 3318, 7727, 17978, 41810, 97214, 226014, 525439, 1221519, 2839710, 6601549, 15346765, 35676927, 82938821, 192809396, 448227496, 1042002541, 2422362052, 5631308596, 13091204252, 30433357644, 70748973053

Offset: 0

Richard Choulet, Apr 05 2008

Previous name: Transform of A000292 without the initial 0 by the T_{0,0} transformation (see link).

Partial sums of A137229. - R. J. Mathar, Nov 04 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Richard Choulet, Curtz-like transformation.
Index entries for linear recurrences with constant coefficients, signature (5,-9,8,-4,1).

Cf. A095263, A136302, A136303, A136304, A136305.

I:=[1,5,16,43,107]; [n le 5 select I[n] else 5*Self(n-1) -9*Self(n-2) +8*Self(n-3) -4*Self(n-4) +Self(n-5): n in [1..41]]; // G. C. Greubel, Apr 19 2021

LinearRecurrence[{5,-9,8,-4,1}, {1,5,16,43,107}, 41] (* G. C. Greubel, Apr 19 2021 *)
CoefficientList[Series[1/((1-x)^2(1-3x+2x^2-x^3)),{x,0,30}],x] (* Harvey P. Dale, Jun 07 2021 *)

@CachedFunction
def A095263(n): return sum(binomial(n+j+2, 3*j+2) for j in (0..n//2))
def A137234(n): return -(n+3) + sum( (-1)^j*(4-j)*A095263(n-j) for j in (0..2))
[A137234(n) for n in (0..40)] # G. C. Greubel, Apr 19 2021

O.g.f: 1/((1-z)^2*(1 - 3*z + 2*z^2 - z^3)).

a(n) = -(n+3) + Sum_{j=0..2} (-1)^j*(4-j)*A095263(n-j). - G. C. Greubel, Apr 19 2021

A137249 Expansion of g.f. z(2-2z+z^2+z^3)/((1+z)(1-3z+2*z^2-z^3)).

Original entry on oeis.org

2, 2, 7, 15, 37, 84, 197, 456, 1062, 2467, 5737, 13335, 31002, 72069, 167542, 389486, 905447, 2104907, 4893317, 11375580, 26445017, 61477204, 142917162, 332242091, 772369157, 1795540447, 4174125122, 9703663625, 22558281082

Offset: 1

Richard Choulet, Apr 05 2008

Previous name was: Transform of A033999 by the T_{0,1} transformation (see link).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Richard Choulet, Curtz-like transformation.
Index entries for linear recurrences with constant coefficients, signature (2,1,-1,1).

Cf. A095263, A136302, A136303, A136304, A136305.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (2-2*x+x^2+x^3)/((1+x)*(1-3*x+2*x^2-x^3)) )); // G. C. Greubel, Apr 11 2021

m:= 40;
S:= series( x*(2-2*x+x^2+x^3)/((1+x)*(1-3*x+2*x^2-x^3)), x, m+1);
seq(coeff(S, x, j), j = 1..m); # G. C. Greubel, Apr 11 2021

LinearRecurrence[{2,1,-1,1},{2,2,7,15},30] (* Harvey P. Dale, Feb 02 2012 *)

def A132749_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (2-2*x+x^2+x^3)/((1+x)*(1-3*x+2*x^2-x^3)) ).list()
A132749_list(40) # G. C. Greubel, Apr 11 2021

O.g.f: z*(2 -2*z +z^2 +z^3)/( (1+z)*(1-3*z+2*z^2-z^3) ).
a(n+4) = 2*a(n+3) + a(n+2) - a(n+1) + a(n).
From G. C. Greubel, Apr 11 2021: (Start)
a(n) = (4*(-1)^n + 10*A095263(n) - 12*A095263(n-1) + 11*A095263(n-2))/7.
a(n) = (1/7)*( 4*(-1)^n + Sum_{j=0..floor(n/2)} ( 10*binomial(n+j+2, 3*j+2) - 12*binomial(n+j+1, 3*j+2) + 11*binomial(n+j, 3*j+2) ) ). (End)

New name using g.f. from Joerg Arndt, Apr 19 2021

A369794 Expansion of 1/(1 - x^5/(1-x)^6).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 6, 21, 56, 126, 253, 474, 870, 1651, 3367, 7372, 16762, 38183, 85290, 185573, 394555, 826752, 1724816, 3613968, 7642004, 16313856, 35052905, 75487110, 162349105, 348018300, 743376838, 1583718457, 3370144462, 7173308802, 15285181447

Offset: 0

Seiichi Manyama, Feb 01 2024

Number of compositions of 6*n-5 into parts 5 and 6.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,7,-1).

Cf. A095263, A290998, A368475.

Cf. A000389, A099242, A017837, A107025.

my(N=40, x='x+O('x^N)); Vec(1/(1-x^5/(1-x)^6))

a(n) = A107025(n)-A107025(n-1). First differences of A107025.
a(n) = A017837(6*n-5) = Sum_{k=0..floor((6*n-5)/5)} binomial(k,6*n-5-5*k) for n > 0.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 7*a(n-5) - a(n-6) for n > 6.
a(n) = Sum_{k=0..floor(n/5)} binomial(n-1+k,n-5*k).

A096261 Number of n-tuples of 0,1,2,3,4,5,6,7,8,9 without consecutive digits.

Original entry on oeis.org

1, 10, 91, 828, 7534, 68552, 623756, 5675568, 51642104, 469892512, 4275561136, 38903414208, 353982925023, 3220897542254, 29307009588171, 266665052127080, 2426390512890816, 22077774624328776, 200886102122914612

Offset: 0

Seppo Mustonen, Aug 01 2004

Sketch of a proof for a general base b >= 2: Let a(n) be the number of n-tuples of 0, 1, ..., b-1 without consecutive digits and s(n,i) the number of them with i (i = 0, 1, ..., b-1) as the last digit. Then it is clear that s(n,i) = a(n-1) - s(n-1, i-1) since when extending a valid n-1-tuple with i those ending with i-1 are not valid as n-tuples.

Thus s(n,0) = a(n-1), s(n,1) = a(n-1) - s(n-1,0) = a(n-1) - a(n-2) and in general s(n,i) = a(n-1) - a(n-2) + a(n-3) - ... + (-1)^i*a(n-i-1), i = 0, 1, ..., b-1. Since a(n) = Sum_{j=0..b-1} s(n,j), we get the recursion formula a(n) = b*a(n-1) -( b-1)*a(n-2) + (b-2)*a(n-3) - ... + (-1)^(b-1)*a(n-b).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9,8,-7,6,-5,4,-3,2,-1).

Case b=3 is A095263.

Column k=10 of A277666.

R:=PowerSeriesRing(Integers(), 30);
Coefficients(R!( 1/(1-10*x+9*x^2-8*x^3+7*x^4-6*x^5+5*x^6-4*x^7+3*x^8-2*x^9+x^10) )); // G. C. Greubel, Apr 17 2021

A096261:=proc(n,b::nonnegint) local s,i; option remember; if n<0 then RETURN(0) fi; if n=0 then RETURN(1) fi; s:=0; for i from 1 to b do s:=s+(-1)^(i-1)*(b-i+1)*A096261(n-i,b); od; end; seq(A096261(i,10),i=0..20);

a[n_]:= a[n]= If[n<0, 0, If[n==0, 1, 10a[n-1] -9a[n-2] +8a[n-3] -7a[n-4] +6a[n-5] -5a[n-6] +4a[n-7] -3a[n-8] +2a[n-9] -a[n-10] ]]; Table[ a[n], {n,0,25}] (* Robert G. Wilson v, Aug 02 2004 *)
LinearRecurrence[{10,-9,8,-7,6,-5,4,-3,2,-1}, {1,10,91,828,7534,68552,623756, 5675568,51642104,469892512}, 30] (* Harvey P. Dale, Dec 16 2013 *)

def A096261_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-10*x+9*x^2-8*x^3+7*x^4-6*x^5+5*x^6-4*x^7+3*x^8-2*x^9+x^10) ).list()
A096261_list(30) # G. C. Greubel, Apr 17 2021

a(n) = 10*a(n-1) - 9*a(n-2) + 8*a(n-3) - 7*a(n-4) + 6*a(n-5) - 5*a(n-6) + 4*a(n-7) - 3*a(n-8) + 2*a(n-9) - 1*a(n-10), a(0)=1, a(n)=0, for n<0.

G.f.: 1/(1 - 10*x + 9*x^2 - 8*x^3 + 7*x^4 - 6*x^5 + 5*x^6 - 4*x^7 + 3*x^8 - 2*x^9 + x^10). - Colin Barker, Dec 06 2012

More terms from Robert G. Wilson v, Aug 02 2004

A127895 Riordan array (1/(1+x)^3, x/(1+x)^3).

Original entry on oeis.org

1, -3, 1, 6, -6, 1, -10, 21, -9, 1, 15, -56, 45, -12, 1, -21, 126, -165, 78, -15, 1, 28, -252, 495, -364, 120, -18, 1, -36, 462, -1287, 1365, -680, 171, -21, 1, 45, -792, 3003, -4368, 3060, -1140, 231, -24, 1, -55, 1287, -6435, 12376, -11628, 5985, -1771, 300, -27, 1

Offset: 0

Paul Barry, Feb 04 2007

The matrix inverse of the convolution triangle of A001764 (number of ternary trees). - Peter Luschny, Oct 09 2022

			Triangle begins
    1;
   -3,     1;
    6,    -6,     1;
  -10,    21,    -9,      1;
   15,   -56,    45,    -12,      1;
  -21,   126,  -165,     78,    -15,      1;
   28,  -252,   495,   -364,    120,    -18,     1;
  -36,   462, -1287,   1365,   -680,    171,   -21,     1;
   45,  -792,  3003,  -4368,   3060,  -1140,   231,   -24,   1;
  -55,  1287, -6435,  12376, -11628,   5985, -1771,   300, -27,   1;
   66, -2002, 12870, -31824,  38760, -26334, 10626, -2600, 378, -30, 1;

G. C. Greubel, Rows n=0..100 of triangle, flattened

Inverse is A127898.
Alternating sign version of A127893.
Cf. A001764, A095263, A127896.

[(-1)^(n-k)*Binomial(n+2*k+2, n-k): k in [0..n], n in [0..10]]; // G. C. Greubel, Apr 29 2018

# Uses function InvPMatrix from A357585. Adds column 1, 0, 0, ... to the left.
InvPMatrix(10, n -> binomial(3*n, n)/(2*n+1)); # Peter Luschny, Oct 09 2022

Table[(-1)^(n-k)*Binomial[n+2*k+2, n-k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 29 2018 *)

for(n=0, 10, for(k=0,n, print1((-1)^(n-k)*binomial(n+2*k+2, n-k), ", "))) \\ G. C. Greubel, Apr 29 2018

flatten([[(-1)^(n-k)*binomial(n+2*k+2, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 16 2021

T(n, k) = (-1)^(n-k)*binomial(n +2*k +2, n-k).
Sum_{k=0..n} T(n, k) = A127896(n) (row sums).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A095263(n) (diagonal sums).

Terms a(50) onward added by G. C. Greubel, Apr 29 2018

cp's OEIS Frontend

A136302 Transform of A000027 by the T_{1,1} transformation (see link).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

Extensions

A136303 Expansion of g.f. (1 +x^2)/((1-x)^2*(1 -3*x +2*x^2 -x^3)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

A137229 Expansion of g.f. x/((1-x)*(1-3*x+2*x^2-x^3)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

Extensions

A368475 Expansion of o.g.f. (1-x)^5/((1-x)^5 - x^4).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A127893 Riordan array (1/(1-x)^3, x/(1-x)^3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A137234 Expansion of g.f. 1/((1-x)^2*(1 - 3*x + 2*x^2 - x^3)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Sage

A136303 Expansion of g.f. (1 +x^2)/((1-x)^2(1 -3x +2*x^2 -x^3)).

A137229 Expansion of g.f. x/((1-x)(1-3x+2*x^2-x^3)).

A137234 Expansion of g.f. 1/((1-x)^2(1 - 3x + 2*x^2 - x^3)).

A137249 Expansion of g.f. z(2-2z+z^2+z^3)/((1+z)(1-3z+2*z^2-z^3)).