cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A115129 Partial sums of A005587. Fourth column of triangle A115127.

Original entry on oeis.org

14, 56, 146, 311, 586, 1015, 1652, 2562, 3822, 5522, 7766, 10673, 14378, 19033, 24808, 31892, 40494, 50844, 63194, 77819, 95018, 115115, 138460, 165430, 196430, 231894, 272286, 318101, 369866, 428141, 493520, 566632, 648142, 738752, 839202
Offset: 0

Views

Author

Wolfdieter Lang, Jan 13 2006

Keywords

Links

Crossrefs

Cf. A024191 (third column of A115127).

Programs

  • Mathematica
    LinearRecurrence[{6,-15,20,-15,6,-1},{14,56,146,311,586,1015},40] (* or *) CoefficientList[Series[(14-28x+20x^2-5x^3)/(1-x)^6,{x,0,40}],x] (* Harvey P. Dale, Apr 24 2016 *)

Formula

G.f.: (14 - 28*x + 20*x^2 - 5*x^3)/(1 - x)^6.
a(n)=A115127(n+4, 4), n>=1.
a(0)=14, a(1)=56, a(2)=146, a(3)=311, a(4)=586, a(5)=1015, a(n)= 6*a(n-1)- 15*a(n-2)+ 20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Harvey P. Dale, Apr 24 2016

A115128 Row sums of triangle A115127.

Original entry on oeis.org

3, 13, 45, 148, 486, 1618, 5478, 18841, 65692, 231713, 825386, 2964937, 10728241, 39065505, 143047469, 526399048, 1945668327, 7220164256, 26889574045, 100469991084, 376513308113, 1414840405426, 5329942273203, 20125253091226
Offset: 2

Views

Author

Wolfdieter Lang, Jan 13 2006

Keywords

Formula

a(n)=sum(A115127(n, m), m=1..), n>=1.

A115134 Third diagonal sequence of triangle A115127.

Original entry on oeis.org

10, 30, 95, 311, 1043, 3564, 12363, 43420, 154088, 551684, 1990326, 7228417, 26405725, 96961560, 357688755, 1324982160, 4926502020, 18379785300, 68783559810, 258141170430, 971310800694, 3663524817768, 13848466284350
Offset: 4

Views

Author

Wolfdieter Lang, Jan 13 2006

Keywords

Crossrefs

Cf. A071726 (second diagonal of A115127 for n>=3 with g.f. -(1+x+3*x^2)+(1+x^2)*c(x)).

Formula

G.f. -(1+x^2+4*x^3) + (1-x+x^3)*c(x) with the g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan).

A024191 [ (3rd elementary symmetric function of 3,4,...,n+4)/(3+4+...+n+4) ].

Original entry on oeis.org

5, 19, 47, 95, 170, 280, 434, 642, 915, 1265, 1705, 2249, 2912, 3710, 4660, 5780, 7089, 8607, 10355, 12355, 14630, 17204, 20102, 23350, 26975, 31005, 35469, 40397, 45820, 51770, 58280, 65384, 73117, 81515, 90615, 100455, 111074, 122512, 134810, 148010
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A115127.
Partial sums of A005586.

Programs

  • Mathematica
    Table[n(n+1)(n^2+13n+46)/24,{n,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{5,19,47,95,170},40] (* Harvey P. Dale, Apr 28 2014 *)
CoefficientList[Series[(5 - 6 x + 2 x^2)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 28 2014 *)
  • PARI
    a(n) = n*(n+1)*(n^2+13*n+46)/24 \\ Charles R Greathouse IV, Oct 21 2022

Formula

a(n)=A115127(n+2, 3), n>=2.
a(n) = n*(n+1)*(n^2+13n+46)/24 =a(n-1)+A005586(n). - Henry Bottomley, Oct 25 2001
G.f.: x*(5-6*x+2*x^2)/(1-x)^5.
a(n) = floor(A024184(n)/A055998(n+2)). - R. J. Mathar, Sep 15 2009
a(1)=5, a(2)=19, a(3)=47, a(4)=95, a(5)=170, a(n)=5*a(n-1)- 10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5). - Harvey P. Dale, Apr 28 2014

A115130 Partial sums of A005557.

Original entry on oeis.org

42, 174, 471, 1043, 2044, 3682, 6230, 10038, 15546, 23298, 33957, 48321, 67340, 92134, 124012, 164492, 215322, 278502, 356307, 451311, 566412, 704858, 870274, 1066690, 1298570, 1570842, 1888929, 2258781, 2686908, 3180414, 3747032
Offset: 0

Views

Author

Wolfdieter Lang, Jan 13 2006

Keywords

Links

Crossrefs

Cf. A115129 (fourth column of A115127).

Programs

  • Mathematica
    Accumulate[LinearRecurrence[{6,-15,20,-15,6,-1},{42,132,297,572,1001,1638},40]] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{42,174,471,1043,2044,3682,6230},40] (* Harvey P. Dale, Feb 22 2024 *)

Formula

G.f.: (42-120*x+135*x^2-70*x^3+14*x^4)/(1-x)^7.
a(n)=A115127(n+5, 5), n>=1.

A115132 Partial sums of A064059.

Original entry on oeis.org

132, 561, 1562, 3564, 7204, 13392, 23388, 38892, 62148, 96063, 144342, 211640, 303732, 427702, 592152, 807432, 1085892, 1442157, 1893426, 2459796, 3164612, 4034844, 5101492, 6400020, 7970820, 9859707, 12118446, 14805312, 17985684
Offset: 0

Views

Author

Wolfdieter Lang, Jan 13 2006

Keywords

Links

Crossrefs

Cf. A115130 (fifth column of A115127).

Programs

  • Mathematica
    LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{132,561,1562,3564,7204,13392,23388,38892},40] (* Harvey P. Dale, Jul 22 2024 *)

Formula

G.f.:(132-495*x+770*x^2-616*x^3+252*x^4-42*x^5)/(1-x)^8.
a(n)=A115127(n+6, 6), n>=1, a(0):=C(6)=132, with C(n):=A000108(n) (Catalan).

A115133 Partial sums of A064061.

Original entry on oeis.org

429, 1859, 5291, 12363, 25623, 48879, 87639, 149655, 245586, 389796, 601304, 904904, 1332474, 1924494, 2731794, 3817554, 5259579, 7152873, 9612537, 12777017, 16811729, 21913089, 28312977, 36283665, 46143240, 58261554
Offset: 0

Views

Author

Wolfdieter Lang, Jan 13 2006

Keywords

Links

Crossrefs

Cf. A114426 (sixth column of A115127).

Programs

  • Mathematica
    Accumulate[Table[Binomial[n,7]-Binomial[n,5],{n,13,50}]] (* or *) LinearRecurrence[ {9,-36,84,-126,126,-84,36,-9,1},{429,1859,5291,12363,25623,48879,87639,149655,245586},40] (* Harvey P. Dale, Sep 03 2015 *)

Formula

G.f.:(429-2002*x+4004*x^2-4368*x^3+2730*x^4-924*x^5+132*x^6)/(1-x)^9.
a(n)=A115127(n+7, 7), n>=1, a(0):=C(7)=429, with C(n):=A000108(n) (Catalan).
Showing 1-7 of 7 results.

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