A084626 -id:A084626 - cp's OEIS frontend

A084631 a(n) = floor(C(n+8,8)/C(n+2,2)).

1, 3, 7, 16, 33, 61, 107, 178, 286, 442, 663, 969, 1384, 1938, 2664, 3605, 4807, 6325, 8222, 10571, 13455, 16965, 21206, 26295, 32364, 39556, 48032, 57970, 69564, 83028, 98595, 116522, 137085, 160585, 187349, 217730, 252109, 290895, 334529, 383484

Offset: 0

Paul Barry, Jun 01 2003

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A084626.

[Floor(Binomial(n+8, 6)/28): n in [0..50]]; // G. C. Greubel, Mar 23 2023

Floor[Binomial[Range[0,50] +8, 6]/28] (* G. C. Greubel, Mar 23 2023 *)

[binomial(n+8,6)//28 for n in range(51)] # G. C. Greubel, Mar 23 2023

a(n) = 1 + floor(n*(n+11)*(n^4 +22*n^3 +203*n^2 +902*n +2232)/20160).

A084627 a(n) = floor(C(n+6,6)/C(n+3,3)).

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 11, 14, 18, 22, 28, 34, 40, 48, 57, 66, 77, 88, 101, 115, 130, 146, 163, 182, 203, 224, 248, 272, 299, 327, 357, 388, 421, 456, 494, 533, 574, 617, 662, 709, 759, 810, 864, 921, 980, 1041, 1105, 1171, 1240, 1311, 1386, 1463, 1542, 1625, 1711

Offset: 0

Paul Barry, Jun 01 2003

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A084626, A084628, A084630, A084631.

[Floor(Binomial(n+6,3)/20): n in [0..70]]; // G. C. Greubel, Mar 24 2023

Table[Floor[Binomial[n+6,6]/Binomial[n+3,3]],{n,0,60}] (* Harvey P. Dale, Feb 07 2015 *)
Floor[Binomial[Range[6,76],3]/20] (* G. C. Greubel, Mar 24 2023 *)

[binomial(n+6,3)//20 for n in range(71)] # G. C. Greubel, Mar 24 2023

a(n) = 1 + floor(n*(n^2 +15*n +74)/120).

a(n) = floor(binomial(n+6,3)/20). - G. C. Greubel, Mar 24 2023

A084624 a(n) = floor(C(n+5,5)/C(n+2,2)).

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 16, 22, 28, 36, 45, 56, 68, 81, 96, 114, 133, 154, 177, 202, 230, 260, 292, 327, 365, 406, 449, 496, 545, 598, 654, 714, 777, 843, 913, 988, 1066, 1148, 1234, 1324, 1419, 1518, 1621, 1729, 1842, 1960, 2082, 2210, 2342, 2480, 2623, 2772, 2926

Offset: 0

Paul Barry, Jun 01 2003

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-3,3,-1).

Cf. A011865, A084625, A084626, A084627, A084628, A084630, A084631.

[Floor(Binomial(n+5,3)/10): n in [0..60]]; // G. C. Greubel, Mar 24 2023

LinearRecurrence[{3,-3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-3,3,-1},{1,2,3, 5,8,12,16,22,28,36,45,56,68,81,96,114,133,154,177,202,230, 260,292},53] (* Ray Chandler, Jul 17 2015 *)
Table[Floor[Binomial[n+5,5]/Binomial[n+2,2]],{n,0,60}] (* or *) Table[ Floor[((3+n)(4+n)(5+n))/60],{n,0,60}] (* Harvey P. Dale, Sep 04 2017 *)
Floor[Binomial[Range[5,65],3]/10] (* G. C. Greubel, Mar 24 2023 *)

[(binomial(n+5,3)//10) for n in range(61)] # G. C. Greubel, Mar 24 2023

a(n) = 1 + floor( n*(n^2 + 12*n + 47)/60 ).
From G. C. Greubel, Mar 24 2023: (Start)
a(n) = floor( binomial(n+5,3)/10 ).
G.f.: (1 -x +x^3 -x^6 +2*x^7 -2*x^8 +2*x^9 -x^10 +x^11 -x^12 +x^14 +x^15 -2*x^16 +x^17)/((1-x)^3*(1-x^20)). (End)

A084625 Binomial transform of A084624.

Original entry on oeis.org

1, 3, 8, 21, 55, 143, 366, 919, 2265, 5491, 13125, 31000, 72485, 168042, 386709, 884161, 2009742, 4543830, 10222264, 22891099, 51041560, 113359224, 250839510, 553173006, 1216070081, 2665518207, 5826533103, 12703217438, 27628250142

Offset: 0

Paul Barry, Jun 01 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A084624, A084626, A084627, A084628, A084630, A084631.

A084625:= func< n | (&+[Binomial(n,j)*Floor(Binomial(j+5,3)/10): j in [0..n]]) >;
[A084625(n): n in [0..50]]; // G. C. Greubel, Mar 24 2023

a[n_]:= a[n]= 2^n +Sum[Binomial[n,j]*Floor[j*(j^2+12*j+47)/60], {j,0, n}];
Table[a[n], {n,0,50}] (* G. C. Greubel, Mar 24 2023 *)

def A084625(n): return sum(binomial(n,j)*(binomial(j+5,3)//10) for j in range(n+1))
[A084625(n) for n in range(51)] # G. C. Greubel, Mar 24 2023

a(n) = Sum_{k=0..n} C(n, k)*floor(C(k+5, 5)/C(k+2, 2)).

a(n) = 2^n + Sum_{k=0..n} binomial(n,k)*floor(k*(k^2 +12*k +47)/60). - G. C. Greubel, Mar 24 2023

cp's OEIS Frontend

A084631 a(n) = floor(C(n+8,8)/C(n+2,2)).

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A084627 a(n) = floor(C(n+6,6)/C(n+3,3)).

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A084624 a(n) = floor(C(n+5,5)/C(n+2,2)).

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A084625 Binomial transform of A084624.

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