A081294 -id:A081294 - cp's OEIS frontend

A345908 Traces of the matrices (A345197) counting integer compositions by length and alternating sum.

1, 1, 0, 1, 3, 3, 6, 15, 24, 43, 92, 171, 315, 629, 1218, 2313, 4523, 8835, 17076, 33299, 65169

Offset: 0

Gus Wiseman, Jul 26 2021

The matrices (A345197) count the integer compositions of n of length k with alternating sum i, where 1 <= k <= n, and i ranges from -n + 2 to n in steps of 2. Here, the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. So a(n) is the number of compositions of n of length (n + s)/2, where s is the alternating sum of the composition.

			The a(0) = 1 through a(7) = 15 compositions of n = 0..7 of length (n + s)/2 where s = alternating sum (empty column indicated by dot):
  ()  (1)  .  (2,1)  (2,2)    (2,3)    (2,4)      (2,5)
                     (1,1,2)  (1,2,2)  (1,3,2)    (1,4,2)
                     (2,1,1)  (2,2,1)  (2,3,1)    (2,4,1)
                                       (1,1,3,1)  (1,1,3,2)
                                       (2,1,2,1)  (1,2,3,1)
                                       (3,1,1,1)  (2,1,2,2)
                                                  (2,2,2,1)
                                                  (3,1,1,2)
                                                  (3,2,1,1)
                                                  (1,1,1,1,3)
                                                  (1,1,2,1,2)
                                                  (1,1,3,1,1)
                                                  (2,1,1,1,2)
                                                  (2,1,2,1,1)
                                                  (3,1,1,1,1)

Traces of the matrices given by A345197.
Diagonals and antidiagonals of the same matrices are A346632 and A345907.
Row sums of A346632.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
Other diagonals are A008277 of A318393 and A055884 of A320808.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000346, A001405, A007318, A008549, A025047, A114121, A163493.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[#]==(n+ats[#])/2&]],{n,0,15}]

A028403 Number of types of Boolean functions of n variables under a certain group.

Original entry on oeis.org

4, 12, 40, 144, 544, 2112, 8320, 33024, 131584, 525312, 2099200, 8392704, 33562624, 134234112, 536903680, 2147549184, 8590065664, 34360000512, 137439477760, 549756862464, 2199025352704, 8796097216512, 35184380477440, 140737505132544, 562949986975744

Offset: 1

N. J. A. Sloane

Harvey P. Dale, Table of n, a(n) for n = 1..1000
I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.
Index entries for sequences related to Boolean functions
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A000051, A000079, A004171, A007582, A081294.

This sequence in base 2 is A163450. - Jaroslav Krizek, Jul 27 2009

[2^(2*n-1) +2^n: n in [1..30]]; // G. C. Greubel, Jul 07 2021

Join[{4},Table[FromDigits[Join[{1},PadRight[{},n-2,0],{1},PadRight[ {},n,0]],2],{n,2,30}]] (* Harvey P. Dale, Jan 24 2021 *)

Vec(4*x*(1-3*x)/((1-2*x)*(1-4*x)) + O(x^100)) \\ Colin Barker, Sep 30 2014

[2^(2*n-1) +2^n for n in (1..30)] # G. C. Greubel, Jul 07 2021

a(n) = (2^(n-1) + 1)*2^n = 2*A007582(n-1). - Ralf Stephan, Mar 24 2004
a(n) = A000079(n) * (A000079(n-1) + 1) = (A000051(n) - 1) * A000051(n-1) = A000079(n) * A000051(n-1) = (A000051(n) - 1) * (A000079(n-1) + 1) = 2^n * (2^(n-1) + 1). a(n+1) = A000079(n+1) * (A000079(n) + 1) = (A000051(n+1) - 1) * A000051(n) = A000079(n+1) * A000051(n) = (A000051(n+1) - 1) * (A000079(n) + 1) = 2^(n+1) * (2^n + 1). a(n) = A081294(n) + A000079(n) = A004171(n-1) + A000079(n) = 2^(2n-1) + 2^n. - Jaroslav Krizek, Jul 27 2009
From Colin Barker, Sep 30 2014: (Start)
a(n) = 6*a(n-1) - 8*a(n-2).
G.f.: 4*x*(1 - 3*x)/((1-2*x)*(1-4*x)). (End)
E.g.f.: (1/2)*(exp(2*x) -1)*(exp(2*x) + 3). - G. C. Greubel, Jul 07 2021

More terms from Vladeta Jovovic, Feb 24 2000

More terms from Colin Barker, Sep 30 2014

A090965 a(n) = 8a(n-1) - 4a(n-2), where a(0) = 1, a(1) = 4.

Original entry on oeis.org

1, 4, 28, 208, 1552, 11584, 86464, 645376, 4817152, 35955712, 268377088, 2003193856, 14952042496, 111603564544, 833020346368, 6217748512768, 46409906716672, 346408259682304, 2585626450591744, 19299378566004736

Offset: 0

Philippe Deléham, Feb 29 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-4).

Cf. A001075.

Sum_{k>=0} A086645(n,k)*m^k for m = 0, 1, 2, 4 gives A000007, A081294, A001541, A083884.

a:=[1,4];; for n in [3..20] do a[n]:=8*a[n-1]-4*a[n-2]; od; a; # G. C. Greubel, Feb 03 2019

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-4*x)/(1-8*x+4*x^2) )); // G. C. Greubel, Feb 03 2019

LinearRecurrence[{8,-4}, {1,4}, 20] (* G. C. Greubel, Feb 03 2019 *)

my(x='x+O('x^20)); Vec((1-4*x)/(1-8*x+4*x^2)) \\ G. C. Greubel, Feb 03 2019

[lucas_number2(n,8,4)/2 for n in range(0,21)] # Zerinvary Lajos, Jul 08 2008

a(n) = Sum_{k>=0} binomial(2*n, 2*k)*3^k = Sum_{k>=0} A086645(n, k)*3^k.
a(n) = 2^n*A001075(n).
G.f.: (1-4*x)/(1-8*x+4*x^2). - Philippe Deléham, Sep 07 2009
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(3*k-4)/(x*(3*k-1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 28 2013
From Peter Bala, Feb 19 2022: (Start)
a(n) = Sum_{k = 0..floor(n/2)} 4^(n-2*k)*12^k*binomial(n,2*k).
a(n) = [x^n] (4*x + sqrt(1 + 12*x^2))^n.
G.f.: A(x) = 1 + x*B'(x)/B(x), where B(x) = 1/sqrt(1 - 8*x + 4*x^2) is the g.f. of A069835.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. (End)

Corrected by T. D. Noe, Nov 07 2006

A136158 Triangle whose rows are generated by A136157^n * [1, 1, 0, 0, 0, ...].

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 9, 15, 7, 1, 27, 54, 36, 10, 1, 81, 189, 162, 66, 13, 1, 243, 648, 675, 360, 105, 16, 1, 729, 2187, 2673, 1755, 675, 153, 19, 1, 2187, 7290, 10206, 7938, 3780, 1134, 210, 22, 1, 6561, 24057, 37908, 34020, 19278, 7182, 1764, 276, 25, 1

Offset: 0

Gary W. Adamson, Dec 16 2007

Triangle T(n,k), 0 <= k <= n, read by rows given by [1,2,0,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 17 2007

Equals A080419 when first column is removed (here). - Georg Fischer, Jul 25 2023

			First few rows of the triangle:
    1;
    1,    1;
    3,    4,    1;
    9,   15,    7,    1;
   27,   54,   36,   10,   1;
   81,  189,  162,   66,  13,   1;
  243,  648,  675,  360, 105,  16,  1;
  729, 2187, 2673, 1755, 675, 153, 19, 1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000007, A000012, A001519, A003688, A006234, A016777, A062741.
Cf. A080419, A080421, A080422, A080423, A081294, A133494, A136157.
Absolute value of A164948.

A136158:= func< n,k | n eq 0 select 1 else 3^(n-k-1)*(n+2*k)* Binomial(n, k)/n >;
[A136158(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 22 2023; Dec 27 2023

A136158[n_,k_]:= If[n==0, 1, 3^(n-k-1)*(n+2*k)*Binomial[n,k]/n];
Table[A136158[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 22 2023; Dec 27 2023 *)

T(n,k) = if ((n<0) || (k<0), return(0)); if ((n==0) && (k==0), return(1)); if (n==1, if (k<=1, return(1))); 3*T(n-1,k) + T(n-1,k-1);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 25 2023

def A136158(n,k): return 1 if (n==0) else 3^(n-k-1)*((n+2*k)/n)*binomial(n, k)
flatten([[A136158(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 22 2023; Dec 27 2023

Sum_{k=0..n} T(n, k) = A081294(n).
Given A136157 = M, an infinite lower triangular bidiagonal matrix with (3, 3, 3, ...) in the main diagonal, (1, 1, 1, ...) in the subdiagonal and the rest zeros; rows of A136157 are generated from M^n * [1, 1, 0, 0, 0, ...], given a(0) = 1.
T(n, k) = A038763(n,n-k). - Philippe Deléham, Dec 17 2007
T(n, k) = 3*T(n-1, k) + T(n-1, k-1) for n > 1, T(0,0) = T(1,1) = T(1,0) = 1. - Philippe Deléham, Oct 30 2013
Sum_{k=0..n} T(n, k)*x^k = (1+x)*(3+x)^(n-1), n >= 1. - Philippe Deléham, Oct 30 2013
G.f.: (1-2*x)/(1-3*x-x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Dec 22 2023: (Start)
T(n, 0) = A133494(n).
T(n, 1) = A006234(n+2).
T(n, 2) = A080420(n-2).
T(n, 3) = A080421(n-3).
T(n, 4) = A080422(n-4).
T(n, 5) = A080423(n-5).
T(n, n) = A000012(n).
T(n, n-1) = A016777(n-1).
T(n, n-2) = A062741(n-1).
Sum_{k=0..n} (-1)^k * T(n, k) = 0^n = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A003688(n).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A001519(n). (End)
From G. C. Greubel, Dec 27 2023: (Start)
T(n, k) = 3^(n-k-1)*(n+2*k)*binomial(n,k)/n, for n > 0, with T(0, 0) = 1.
T(n, k) = (-1)^k * A164948(n, k). (End)

More terms from Philippe Deléham, Dec 17 2007

A070832 a(n) = Sum_{k=0..n} binomial(8n,8k).

Original entry on oeis.org

1, 2, 12872, 1470944, 622116992, 125858012672, 36758056208384, 8793364151263232, 2334899414608412672, 586347560750962049024, 151652224498623981289472, 38612725801339748322639872, 9913426188311626771400228864

Offset: 0

Sebastian Gutierrez and Sarah Kolitz (skolitz(AT)mit.edu), May 15 2002

Seiichi Manyama, Table of n, a(n) for n = 0..415
Index entries for linear recurrences with constant coefficients, signature (136,32880,-552704,-65536).

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), A070782 (b=5), A070967 (b=6), A094211 (b=7), this sequence (b=8), A094213 (b=9), A070833 (b=10).

Table[Sum[Binomial[8n,8k],{k,0,n}],{n,0,15}] (* Harvey P. Dale, Nov 25 2020 *)

a(n)=sum(k=0,n,binomial(8*n,8*k)); \\ Benoit Cloitre, May 27 2004

Vec((1 - 134*x - 20280*x^2 + 207296*x^3 + 8192*x^4) / ((1 - 16*x)*(1 - 256*x)*(1 + 136*x + 16*x^2)) + O(x^15)) \\ Colin Barker, May 27 2019

Let b(n) = a(n)-2^(8*n)/8 then b(n)+120*b(n-1)-2160*b(n-2)-256*b(n-3)=0. - Benoit Cloitre, May 27 2004
a(n) = 1/4*16^n + 1/8*256^n + 1/4*(-68 + 48*sqrt(2))^n + 1/4*(-68-48*sqrt(2))^n.
From Colin Barker, May 27 2019: (Start)
G.f.: (1 - 134*x - 20280*x^2 + 207296*x^3 + 8192*x^4) / ((1 - 16*x)*(1 - 256*x)*(1 + 136*x + 16*x^2)).
a(n) = 21*a(n-1) + 353*a(n-2) - 32*a(n-3) for n>4.
(End)

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 15 2007

A070967 a(n) = Sum_{k=0..n} binomial(6n,6k).

Original entry on oeis.org

1, 2, 926, 37130, 2973350, 174174002, 11582386286, 729520967450, 47006639297270, 2999857885752002, 192222214478506046, 12295976362284182570, 787111112023373201990, 50370558298891875954002, 3223838658635388303336206, 206322355109994528871954490

Offset: 0

Sebastian Gutierrez and Sarah Kolitz (skolitz(AT)mit.edu), May 16 2002

Matthijs Coster, Supercongruences, Thesis, Jun 08, 1988.

Seiichi Manyama, Table of n, a(n) for n = 0..554
Index entries for linear recurrences with constant coefficients, signature (38,1691,-1728).

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), A070782 (b=5), this sequence (b=6), A094211 (b=7), A070832 (b=8), A094213 (b=9), A070833 (b=10).

Table[Sum[Binomial[6n,6k],{k,0,n}],{n,0,20}] (* or *) LinearRecurrence[ {38,1691,-1728},{1,2,926,37130},30] (* Harvey P. Dale, Jun 19 2021 *)

PARI
```
a(n)=sum(k=0,n,binomial(6*n,6*k))
```

a(n)=if(n<0,0,(2*(-27)^n+2+64^n+0^n)/6)

a(n)=if(n<0,0,polsym(x*(x-64)*(x+27)^2*(x-1)^2,n)[n+1]/6)

G.f.: (1-36x-841x^2+288x^3)/((1-x)*(1+27x)*(1-64x)).
a(n) = ((-27)^n + 1)/3 + (64^n + 0^n)/6.
Let b(n) = a(n)-2^(6n)/6 then b(n)+26*b(n-1)-27*b(n-2) = 0. - Benoit Cloitre, May 27 2004

A038763 Triangular matrix arising in enumeration of catafusenes, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 7, 15, 9, 1, 10, 36, 54, 27, 1, 13, 66, 162, 189, 81, 1, 16, 105, 360, 675, 648, 243, 1, 19, 153, 675, 1755, 2673, 2187, 729, 1, 22, 210, 1134, 3780, 7938, 10206, 7290, 2187, 1, 25, 276, 1764, 7182, 19278, 34020, 37908, 24057, 6561, 1, 28, 351, 2592, 12474, 40824, 91854, 139968, 137781, 78732, 19683

Offset: 0

N. J. A. Sloane, May 03 2000

Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005
Triangle read by rows, n-th row = X^(n-1) * [1, 1, 0, 0, 0, ...] where X = an infinite bidiagonal matrix with (1,1,1,...) in the main diagonal and (3,3,3,...) in the subdiagonal; given row 0 = 1. - Gary W. Adamson, Jul 19 2008
Fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(2x+1)^n; see A193722 for the definition of fusion of two sequences of polynomials or triangular arrays. - Clark Kimberling, Aug 04 2011

			Triangle begins:
  1;
  1,  1;
  1,  4,   3;
  1,  7,  15,   9;
  1, 10,  36,  54,   27;
  1, 13,  66, 162,  189,   81;
  1, 16, 105, 360,  675,  648,  243;
  1, 19, 153, 675, 1755, 2673, 2187, 729;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.

Cf. A000007, A000012, A001296, A006138, A006234, A016777, A024462.
Cf. A051836, A062741, A080420, A080421, A080422, A080423, A081294.
Cf. A084938, A098399, A110523, A133494, A136158, A193722.

A038763:= func< n,k | n eq 0 select 1 else 3^(k-1)*(3*n-2*k)*Binomial(n,k)/n >;
[A038763(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 27 2023

A038763[n_,k_]:= If[n==0, 1, 3^(k-1)*(3*n-2*k)*Binomial[n,k]/n];
Table[A038763[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 27 2023 *)

T(n,k) = if ((n<0) || (k<0), return(0)); if ((n==0) && (k==0), return(1)); if (n==1, if (k<=1, return(1))); T(n-1,k) + 3*T(n-1,k-1);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", "))); \\ Michel Marcus, Jul 25 2023

def A038763(n,k): return 1 if (n==0) else 3^(k-1)*(3*n-2*k)*binomial(n,k)/n
flatten([[A038763(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 27 2023

T(n, 0)=1; T(1, 1)=1; T(n, k)=0 for k>n; T(n, k) = T(n-1, k-1)*3 + T(n-1, k) for n >= 2.
Sum_{k=0..n} T(n,k) = A081294(n). - Philippe Deléham, Sep 22 2006
T(n, k) = A136158(n, n-k). - Philippe Deléham, Dec 17 2007
G.f.: (1-2*x*y)/(1-(3*y+1)*x). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Dec 27 2023: (Start)
T(n, 0) = A000012(n).
T(n, 1) = A016777(n-1).
T(n, 2) = A062741(n-1).
T(n, 3) = 9*A002411(n-2).
T(n, 4) = 27*A001296(n-3).
T(n, 5) = 81*A051836(n-4).
T(n, n) = A133494(n).
T(n, n-1) = A006234(n+2).
T(n, n-2) = A080420(n-2).
T(n, n-3) = A080421(n-3).
T(n, n-4) = A080422(n-4).
T(n, n-5) = A080423(n-5).
T(2*n, n) = 4*A098399(n-1) + (2/3)*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A006138(n-1) + (2/3)*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A110523(n-1) + (4/3)*[n=0]. (End)

More terms from Michel Marcus, Jul 25 2023

A070833 a(n) = Sum_{k=0..n} binomial(10n,10k).

Original entry on oeis.org

1, 2, 184758, 60090032, 139541849878, 94278969044262, 126648421364527548, 111019250117021378442, 125257104438594491956518, 121088185204450642433930072, 128442558588779813655233443038, 128767440665677943753184267342902

Offset: 0

Sebastian Gutierrez and Sarah Kolitz (skolitz(AT)mit.edu), May 15 2002

Seiichi Manyama, Table of n, a(n) for n = 0..332
Index entries for linear recurrences with constant coefficients, signature (522,587797,-75135226,-392963125,3200000).

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), A070782 (b=5), A070967 (b=6), A094211 (b=7), A070832 (b=8), A094213 (b=9), this sequence (b=10). Cf. A000032.

a(n) = sum(k=0, n, binomial(10*n, 10*k)); \\ Michel Marcus, Mar 15 2019

Vec((1 - 520*x - 404083*x^2 + 37605988*x^3 + 117888625*x^4 - 320000*x^5) / ((1 - 1024*x)*(1 - 123*x + x^2)*(1 + 625*x + 3125*x^2)) + O(x^15)) \\ Colin Barker, Mar 15 2019

a(n) = 1/10*1024^n+1/5*(-625/2+275/2*sqrt(5))^n+1/5*(-625/2-275/2*sqrt(5))^n+1/5*(123/2+55/2*sqrt(5))^n+1/5*(123/2-55/2*sqrt(5))^n.
G.f.: (1 - 520*x - 404083*x^2 + 37605988*x^3 + 117888625*x^4 - 320000*x^5) / ((1 - 1024*x)*(1 - 123*x + x^2)*(1 + 625*x + 3125*x^2)). - Colin Barker, Mar 15 2019
a(2*n) = (2^(20*n-1) + Lucas(20*n) + 5^(5*n)*Lucas(10*n))/5, for n>0 and for Lucas(n) = A000032(n). - Greg Dresden, Feb 04 2023

A094211 a(n) = Sum_{k=0..n} binomial(7n,7k).

Original entry on oeis.org

1, 2, 3434, 232562, 42484682, 4653367842, 644032289258, 79450506979090, 10353832741654602, 1313930226050847938, 168883831255263816554, 21573903987107973878962, 2764126124873404346104778, 353643666623193292098680930, 45276535087893983968685884906

Offset: 0

Benoit Cloitre, May 27 2004

Seiichi Manyama, Table of n, a(n) for n = 0..474
Index entries for linear recurrences with constant coefficients, signature (71,7585,-36991,-128).

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), A070782 (b=5), A070967 (b=6), this sequence (b=7), A070832 (b=8), A094213 (b=9), A070833 (b=10).

A094211:=n->add(binomial(7*n,7*k), k=0..n): seq(A094211(n), n=0..20); # Wesley Ivan Hurt, Feb 16 2017

Table[Sum[Binomial[7n,7k],{k,0,n}],{n,0,20}] (* or *) LinearRecurrence[ {71,7585,-36991,-128},{1,2,3434,232562},20] (* Harvey P. Dale, May 06 2012 *)

PARI
```
a(n)=sum(k=0,n,binomial(7*n,7*k))
```

Let b(n) = a(n) - 2^(7n)/7; then b(n) + 57*b(n-1) - 289*b(n-2) - b(n-3) = 0.
a(n) = 71*a(n-1) + 7585*a(n-2) - 36991*a(n-3) - 128*a(n-4); a(0)=1, a(1)=2, a(2)=3434, a(3)=232562. - Harvey P. Dale, May 06 2012
G.f.: (1 - 69*x - 4293*x^2 + 10569*x^3) / ((1 - 128*x)*(1 + 57*x - 289*x^2 - x^3)). - Colin Barker, May 27 2019
a(n) = (1 + 2*(s*(3 - 4*s^2))^(7*n) + 2*(-1)^n*((1 - 2*s^2)^(7*n) + s^(7*n))) * 2^(7*n)/7, where s = sin(Pi/14). - Vaclav Kotesovec, Apr 17 2023

A094213 a(n) = Sum_{k=0..n} binomial(9n,9k).

Original entry on oeis.org

1, 2, 48622, 9373652, 9263421862, 3433541316152, 2140802758677844, 984101481334553024, 536617781178725122150, 265166261617029717011822, 138567978655457801631498052, 70126939586658252408697345838, 36144812798331420987905742371116

Offset: 0

Benoit Cloitre, May 27 2004

Seiichi Manyama, Table of n, a(n) for n = 0..369
Index entries for linear recurrences with constant coefficients, signature (265,139823,-6826204,-6965249,512).

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), A070782 (b=5), A070967 (b=6), A094211 (b=7), A070832 (b=8), this sequence (b=9), A070833 (b=10).

Table[Sum[Binomial[9n,9k],{k,0,n}],{n,0,15}] (* Harvey P. Dale, Jul 14 2019 *)

PARI
```
a(n)=sum(k=0,n,binomial(9*n,9*k))
```

Vec((1 - 263*x - 91731*x^2 + 3035380*x^3 + 1547833*x^4) / ((1 + x)*(1 - 512*x)*(1 + 246*x - 13605*x^2 + x^3)) + O(x^15)) \\ Colin Barker, May 27 2019

Let b(n) = a(n)-2^(9*n)/9 then b(n)+246*b(n-1)-13605*b(n-2)+b(n-3)+(-1)^n*3078=0.
Conjectures from Colin Barker, May 27 2019: (Start)
G.f.: (1 - 263*x - 91731*x^2 + 3035380*x^3 + 1547833*x^4) / ((1 + x)*(1 - 512*x)*(1 + 246*x - 13605*x^2 + x^3)).
a(n) = 265*a(n-1) + 139823*a(n-2) - 6826204*a(n-3) - 6965249*a(n-4) + 512*a(n-5) for n>4. (End)
a(n) ~ (1/9)*exp(n*9*log(2)) (conjecture). - Bill McEachen, Aug 11 2025

cp's OEIS Frontend

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A090965 a(n) = 8*a(n-1) - 4*a(n-2), where a(0) = 1, a(1) = 4.

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A136158 Triangle whose rows are generated by A136157^n * [1, 1, 0, 0, 0, ...].

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A070832 a(n) = Sum_{k=0..n} binomial(8*n,8*k).

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A070967 a(n) = Sum_{k=0..n} binomial(6*n,6*k).

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A038763 Triangular matrix arising in enumeration of catafusenes, read by rows.

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A090965 a(n) = 8a(n-1) - 4a(n-2), where a(0) = 1, a(1) = 4.

A070832 a(n) = Sum_{k=0..n} binomial(8n,8k).

A070967 a(n) = Sum_{k=0..n} binomial(6n,6k).

A070833 a(n) = Sum_{k=0..n} binomial(10n,10k).

A094211 a(n) = Sum_{k=0..n} binomial(7n,7k).

A094213 a(n) = Sum_{k=0..n} binomial(9n,9k).