A137361 -id:A137361 - cp's OEIS frontend

A137356 a(n) = Sum_{k <= n/2 } binomial(n-2k, 3k).

1, 1, 1, 1, 1, 2, 5, 11, 21, 36, 58, 92, 149, 250, 431, 750, 1299, 2227, 3784, 6401, 10828, 18364, 31236, 53228, 90741, 154603, 263178, 447702, 761403, 1295022, 2203162, 3749001, 6380241, 10858285, 18478155, 31443013, 53501860, 91034937, 154900529, 263576791

Offset: 0

Don Knuth, Apr 11 2008

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Robert Israel, Table of n, a(n) for n = 0..4329
V. C. Harris, C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,2,3).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,1).

Cf. A137357-A137361, A136444, A137402.

I:=[1,1,1,1,1]; [n le 5 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3)+Self(n-5): n in [1..45]]; // Vincenzo Librandi, Aug 09 2015

f:= gfun:-rectoproc({a(n) = 3*a(n-1)-3*a(n-2)+a(n-3)+a(n-5),seq(a(i)=1,i=0..4)},a(n),remember):
map(f, [$0..50]); # Robert Israel, May 26 2017

LinearRecurrence[{3, -3, 1, 0, 1}, {1, 1, 1, 1, 1}, 50] (* Vincenzo Librandi, Aug 09 2015 *)
CoefficientList[Series[(1-x)^2/(1 - 3 x + 3 x^2 - x^3 - x^5), {x, 0, 40}], x] (* Vaclav Kotesovec, Aug 09 2015 *)
Table[Sum[Binomial[n-2k,3k],{k,0,n/2}],{n,0,50}] (* Harvey P. Dale, Nov 07 2021 *)

Let A_n = Sum_{k<=n/2}binomial(n-2k,3k) (the present sequence), B_n= Sum_{k<=n/2}binomial(n-2k, 3k+1)(A137357), C_n= Sum_{k<=n/2}binomial(n-2k, 3k+2) (A137358).
Then A_n = A_{n-1} + C_{n-3} + \delta_{n0}, B_n=B_{n-1} + A_{n-1}, C_n=C_{n-1} + B_{n-1};
so the generating functions are A = (1-z)^2/p(z), B=z(1-z)/p(z), C=z^2/p(z),
where p(z) = (1-z)^3 - z^5 = 1 - 3z + 3z^2 - z^3 - z^5.
The growth ratio is the real root of r^2(r-1)^3 = 1, approximately 1.70161.
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3)+a(n-5). - Vincenzo Librandi, Aug 09 2015
a(n) = hypergeom([-(1/5)*n, -(1/5)*n+1/5, 2/5-(1/5)*n, 3/5-(1/5)*n, -(1/5)*n+4/5], [1/3, 2/3, -(1/2)*n, -(1/2)*n+1/2], -3125/108). - Robert Israel, May 26 2017
G.f.: -(x-1)^2/(-1+3*x-3*x^2+x^3+x^5) . - R. J. Mathar, May 29 2017

A137357 a(n) = Sum_{k <= n/2 } binomial(n-2k, 3k+1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 12, 23, 44, 80, 138, 230, 379, 629, 1060, 1810, 3109, 5336, 9120, 15521, 26349, 44713, 75949, 129177, 219918, 374521, 637699, 1085401, 1846804, 3141826, 5344988, 9093989, 15474230, 26332515, 44810670, 76253683, 129755543, 220790480

Offset: 0

Don Knuth, Apr 11 2008

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

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

Cf. A137356, A137358, A137359, A137360, A137361, A136444.

LinearRecurrence[{3, -3, 1, 0, 1}, Range[0, 4], 50] (* Paolo Xausa, Mar 17 2024 *)

G.f.: x*(x-1) / (x^5+x^3-3*x^2+3*x-1). - Colin Barker, Jan 23 2013

A136444 a(n) = Sum_{k=0..n} kbinomial(n-k, 2k).

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 12, 25, 51, 101, 197, 381, 731, 1392, 2634, 4958, 9290, 17337, 32239, 59760, 110460, 203651, 374593, 687567, 1259597, 2303449, 4205493, 7666560, 13956532, 25374108, 46076436, 83575025, 151431099, 274108826, 495708364, 895670733, 1617003823, 2916984121

Offset: 0

Don Knuth, Apr 04 2008

Consider four related sequences: A_n = sum C(n-k, 2*k), B_n = sum C(n-k, 2*k+1), A^*_n = sum k*C(n-k, 2*k), B^*_n = sum k*C(n-k, 2*k+1).
Sequence A_n, with generating function (1-z)/p(z) where p(z) = 1 - 2*z + z^2 - z^3, is A005251.
Sequence B_n, with generating function z/p(z), is A005314.
Sequence A^*_n is the present sequence.
Sequence B^*_n is A118430, but shifted one place so that the generating function is z^4/p(z)^2 instead of z^3/p(z)^2.
These sequences have many interrelations; for example,
B_{n+1} - B_n = A_n; B^*_{n+1} - B^*_n = A^*_n;
A_{n+1} - A_n = B_{n-1}; A^*{n+1} - A^*_n = B^*{n-1} + B_{n-1}.

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

T. Mansour and M. Shattuck, Counting Peaks and Valleys in a Partition of a Set, J. Int. Seq. 13 (2010), #10.6.8, partitions of [n] with 2 blocks with 1 peak.

Cf. A005251, A005314, A118430, A136445, A137356-A137361.

[&+[k*Binomial(n-k, 2*k): k in [0..n]]: n in [0..40]]; // Bruno Berselli, Feb 13 2015

a:= n-> (Matrix([[0,0,1,1,-3,-5]]). Matrix(6, (i,j)-> if (i=j-1) then 1 elif j=1 then [4,-6,6,-5,2,-1][i] else 0 fi)^n)[1,1]: seq(a(n), n=0..37);  # Alois P. Heinz, Aug 13 2008

a[n_] := ({0, 0, 1, 1, -3, -5} . MatrixPower[ Table[If[i == j-1, 1, If[j == 1, {4, -6, 6, -5, 2, -1}[[i]], 0]], {i, 6}, {j, 6}], n])[[1]]; Table[a[n], {n, 0, 37}] (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz *)
CoefficientList[Series[x^3 (1 - x)/(1 - 2 x + x^2 - x^3)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 15 2015 *)

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

a(n) ~ c * d^n * n, where d = A109134 = 1.75487766624669276... is the root of the equation d*(d-1)^2 = 1, c = 0.072838349685011... is the root of the equation 529*c^3 - 207*c^2 + 26*c = 1. - Vaclav Kotesovec, May 25 2015

A137358 a(n) = Sum_{k <= n/2 } binomial(n-2k, 3k+2).

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 22, 34, 57, 101, 181, 319, 549, 928, 1557, 2617, 4427, 7536, 12872, 21992, 37513, 63862, 108575, 184524, 313701, 533619, 908140, 1545839, 2631240, 4478044, 7619870, 12964858, 22058847, 37533077, 63865592, 108676262, 184929945, 314685488

Offset: 0

Don Knuth, Apr 11 2008

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,1).

Cf. A137356, A137357, A137359, A137360, A137361, A136444.

Table[Sum[Binomial[n-2k,3k+2],{k,0,Floor[n/2]}],{n,0,50}] (* or *) LinearRecurrence[{3,-3,1,0,1},{0,0,1,3,6},50] (* Harvey P. Dale, Nov 06 2012 *)

a(0)=0, a(1)=0, a(2)=1, a(3)=3, a(4)=6, a(n)=3*a(n-1)-3*a(n-2)+ a(n-3)+ a(n-5). - Harvey P. Dale, Nov 06 2012

G.f.: -x^2/(x^5+x^3-3*x^2+3*x-1). - Colin Barker, Jan 23 2013

A137359 a(n) = Sum_{k <= n/2 } k*binomial(n-2k, 3k).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 4, 10, 20, 35, 58, 98, 176, 333, 640, 1213, 2242, 4052, 7226, 12835, 22842, 40788, 72952, 130344, 232200, 412190, 729466, 1288216, 2272012, 4003795, 7050358, 12404345, 21801674, 38275760, 67125420, 117604174, 205865368, 360090917, 629414866

Offset: 0

Don Knuth, Apr 11 2008

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,8,-7,6,-2,0,-1).

Cf. A137356-A137361, A136444.

a:= n-> (Matrix([[10, 4, 1, 0$7]]). Matrix (10, (i,j)-> if i=j-1 then 1 elif j=1 then [6, -15, 20, -15, 8, -7, 6, -2, 0, -1][i] else 0 fi)^n)[1,8]: seq (a(n), n=0..50); # Alois P. Heinz, Oct 23 2008

LinearRecurrence[{6, -15, 20, -15, 8, -7, 6, -2, 0, -1}, {0, 0, 0, 0, 0, 1, 4, 10, 20, 35}, 50] (* Paolo Xausa, Mar 17 2024 *)

G.f.: x^5*(1-x)^2/(x^5+x^3-3*x^2+3*x-1)^2. - Alois P. Heinz, Oct 23 2008

A137360 a(n) = Sum_{k <= n/2 } k*binomial(n-2k, 3k+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 5, 15, 35, 70, 128, 226, 402, 735, 1375, 2588, 4830, 8882, 16108, 28943, 51785, 92573, 165525, 295869, 528069, 940259, 1669725, 2957941, 5229953, 9233748, 16284106, 28688451, 50490125, 88765885, 155891305, 273495479, 479360847, 839451764

Offset: 0

Don Knuth, Apr 11 2008

nonn

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 8, -7, 6, -2, 0, -1).

Cf. A137356-A137361, A136444.

a:= n-> (Matrix([[35, 15, 5, 1, 0$6]]). Matrix (10, (i,j)-> if i=j-1 then 1 elif j=1 then [6, -15, 20, -15, 8, -7, 6, -2, 0, -1][i] else 0 fi)^n)[1,10]: seq (a(n), n=0..50);  # Alois P. Heinz, Oct 23 2008

Table[Sum[k Binomial[n-2k,3k+1],{k,n/2}],{n,0,40}] (* or *) LinearRecurrence[ {6,-15,20,-15,8,-7,6,-2,0,-1},{0,0,0,0,0,0,1,5,15,35},40] (* Harvey P. Dale, May 31 2017 *)

G.f.: x^6*(1-x)/(x^5+x^3-3*x^2+3*x-1)^2. - Alois P. Heinz, Oct 23 2008

A138780 Triangle read by rows: T(n,k)=k*binomial(n-2k,3k+2) (n>=7, 1<=k<=(n-2)/5).

Original entry on oeis.org

1, 6, 21, 56, 126, 252, 2, 462, 18, 792, 90, 1287, 330, 2002, 990, 3003, 2574, 3, 4368, 6006, 36, 6188, 12870, 234, 8568, 25740, 1092, 11628, 48620, 4095, 15504, 87516, 13104, 4, 20349, 151164, 37128, 60

Offset: 7

Emeric Deutsch, May 10 2008

Row n contains floor((n-2)/5) terms.

Row sums yield A137361.

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Cf. A137361.

T:=proc(n,k) options operator, arrow: k*binomial(n-2*k, 3*k+2) end proc: for n from 7 to 23 do seq(T(n,k),k=1..(n-2)*1/5) end do; # yields sequence in triangular form

Flatten[Table[k*Binomial[n-2k,3k+2],{n,7,30},{k,1,(n-2)/5}]] (* Harvey P. Dale, Aug 22 2015 *)

cp's OEIS Frontend

A137356 a(n) = Sum_{k <= n/2 } binomial(n-2k, 3k).

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A137357 a(n) = Sum_{k <= n/2 } binomial(n-2k, 3k+1).

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A136444 a(n) = Sum_{k=0..n} k*binomial(n-k, 2*k).

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A137358 a(n) = Sum_{k <= n/2 } binomial(n-2k, 3k+2).

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A137359 a(n) = Sum_{k <= n/2 } k*binomial(n-2k, 3k).

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Maple

Mathematica

Formula

A137360 a(n) = Sum_{k <= n/2 } k*binomial(n-2k, 3k+1).

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Formula

A138780 Triangle read by rows: T(n,k)=k*binomial(n-2k,3k+2) (n>=7, 1<=k<=(n-2)/5).

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A136444 a(n) = Sum_{k=0..n} kbinomial(n-k, 2k).