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 1-10 of 15 results. Next

A027178 a(n) = T(n,0) + T(n,1) + ... + T(n,n), T given by A027170.

Original entry on oeis.org

1, 6, 20, 52, 120, 260, 544, 1116, 2264, 4564, 9168, 18380, 36808, 73668, 147392, 294844, 589752, 1179572, 2359216, 4718508, 9437096, 18874276, 37748640, 75497372, 150994840, 301989780, 603979664, 1207959436, 2415918984
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Define a triangle U(n,k) with U(n,0) = n*(n+1) + 1 for n>=0 and U(r,c) = U(r-1,c-1) + U(r-1,c). The sum of the terms in row n is a(n). The first rows are 1; 3, 3; 7, 6, 7; 13, 13, 13, 13; 21, 26, 26, 26, 21; row sums are 1, 6, 20, 52, 120. - J. M. Bergot, Feb 15 2013
This triangle is now A222405. - N. J. A. Sloane, Feb 18 2013

Crossrefs

Cf. A222405.

Formula

a(n) = 9*2^n - 4n - 8 (conjectured). - Ralf Stephan, Feb 13 2004
Conjectures from Colin Barker, Feb 17 2016: (Start)
a(n) = 4*a(n-1)-5*a(n-2)+2*a(n-3) for n>2.
G.f.: (1+x)^2 / ((1-x)^2*(1-2*x)).
(End)

A027175 a(n) = A027170(2n-1, n-1).

Original entry on oeis.org

3, 19, 76, 283, 1046, 3890, 14582, 55051, 209062, 797806, 3056868, 11752674, 45316896, 175175816, 678639146, 2634146411, 10241938406, 39882831446, 155519160716, 607181138846, 2373227900936, 9285480209456, 36364319046896
Offset: 1

Views

Author

Clark Kimberling

Keywords

A027180 a(n) = Sum_{0<=j<=i<=n} A027170(i, j).

Original entry on oeis.org

1, 7, 27, 79, 199, 459, 1003, 2119, 4383, 8947, 18115, 36495, 73303, 146971, 294363, 589207, 1178959, 2358531, 4717747, 9436255, 18873351, 37747627, 75496267, 150993639, 301988479, 603978259, 1207957923, 2415917359, 4831836343, 9663674427, 19327350715
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Partial sums of A027178.

Programs

  • Mathematica
    LinearRecurrence[{5,-9,7,-2},{1,7,27,79},50] (* Harvey P. Dale, Jul 08 2019 *)
  • PARI
    Vec((1+x)^2/((1-x)^3*(1-2*x)) + O(x^40)) \\ Colin Barker, Feb 20 2016

Formula

a(n) = 18*2^n - 2*n^2 - 10*n - 17.
From Colin Barker, Feb 20 2016: (Start)
a(n) = 5*a(n-1)-9*a(n-2)+7*a(n-3)-2*a(n-4) for n>3.
G.f.: (1+x)^2 / ((1-x)^3*(1-2*x)).
(End)

A027183 a(n) = n-th largest even number in array T given by A027170.

Original entry on oeis.org

10, 30, 42, 58, 76, 94, 138, 156, 190, 250, 264, 318, 362, 394, 472, 478, 570, 670, 740, 778, 790, 894, 984, 1018, 1046, 1106, 1150, 1290, 1438, 1594, 1748, 1758, 1790, 1930, 2070, 2096, 2110, 2218, 2298, 2494, 2698, 2900, 2910, 3012, 3130, 3266, 3358, 3594, 3748, 3838, 3890
Offset: 1

Views

Author

Clark Kimberling

Keywords

Extensions

Corrected and extended by Sean A. Irvine, Oct 24 2019

A027171 a(n) = A027170(2n, n).

Original entry on oeis.org

1, 10, 42, 156, 570, 2096, 7784, 29168, 110106, 418128, 1595616, 6113740, 23505352, 90633796, 350351636, 1357278296, 5268292826, 20483876816, 79765662896, 311038321436, 1214362277696, 4746455801876, 18570960418916
Offset: 0

Views

Author

Clark Kimberling

Keywords

A027172 a(n) = (1/2) * A027170(2n, n).

Original entry on oeis.org

5, 21, 78, 285, 1048, 3892, 14584, 55053, 209064, 797808, 3056870, 11752676, 45316898, 175175818, 678639148, 2634146413, 10241938408, 39882831448, 155519160718, 607181138848, 2373227900938, 9285480209458, 36364319046898
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

A027175(n) + 2.

A027173 a(n) = A027170(2n, n-1).

Original entry on oeis.org

5, 30, 123, 472, 1790, 6794, 25879, 98952, 379674, 1461248, 5638930, 21811540, 84542016, 328287506, 1276868111, 4973645576, 19398954626, 75753497816, 296142817406, 1158865623236, 4539024407576, 17793358627976, 69805770185498, 274055019608372, 1076651019788200
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A027170.

Programs

  • Mathematica
    a[n_]:=Binomial[2 n,-1+n]+2 Binomial[2+2 n,n]-4; Array[a,30] (* Stefano Spezia, Sep 02 2025 *)
CoefficientList[Series[(-2 + 2*Sqrt[1-4*x] + 5*x - Sqrt[1-4*x]*x - x^2 + 3*Sqrt[1-4*x]*x^2 - 2*x^3 + 4*Sqrt[1-4*x]*x^3)/(2*Sqrt[1-4*x]*(-1 + x)*x^2),{x,0,25}],x] (* Stefano Spezia, Sep 02 2025 *)
CoefficientList[Series[2 - 4*Exp[x] + 4*Exp[2*x]*BesselI[0, 2*x] + (Exp[2*x]*(5*x - 2)*BesselI[1, 2*x])/x,{x,0,25}],x]*Range[0,25]! (* Stefano Spezia, Sep 02 2025 *)
  • PARI
    my(x='x+O('x^40)); Vec(serlaplace(2 - 4*exp(x) + 4*exp(2*x)*besseli(0, 2*x) + (exp(2*x)*(5*x - 2)*besseli(1, 2*x)))) \\ Michel Marcus, Sep 04 2025

Formula

From Stefano Spezia, Sep 02 2025: (Start)
a(n) = binomial(2*n,n-1) + 2*binomial(2*(1+n),n) - 4.
G.f.: (-2 + 2*sqrt(1-4*x) + 5*x - sqrt(1-4*x)*x - x^2 + 3*sqrt(1-4*x)*x^2 - 2*x^3 + 4*sqrt(1-4*x)*x^3)/(2*sqrt(1-4*x)*(-1 + x)*x^2).
E.g.f.: 2 - 4*exp(x) + 4*exp(2*x)*BesselI(0, 2*x) + (exp(2*x)*(5*x - 2)*BesselI(1, 2*x))/x. (End)

Extensions

a(24)-a(25) from Stefano Spezia, Sep 02 2025

A027174 a(n) = A027170(2n, n-2).

Original entry on oeis.org

9, 58, 264, 1106, 4495, 18014, 71652, 283760, 1120806, 4419928, 17413572, 68569666, 269941451, 1062631046, 4183370636, 16471711736, 64870158866, 255541666976, 1006930883396, 3968854010936, 15648092510618, 61714841143568
Offset: 2

Views

Author

Clark Kimberling

Keywords

A027176 a(n) = A027170(2n-1, n-2).

Original entry on oeis.org

7, 43, 185, 740, 2900, 11293, 43897, 170608, 663438, 2582058, 10058862, 39225116, 153111686, 598228961, 2339499161, 9157016216, 35870666366, 140623656686, 551684484386, 2165796506636, 8507878418516, 33441451138598
Offset: 2

Views

Author

Clark Kimberling

Keywords

A027177 a(n) = A027170(n, floor(n/2)).

Original entry on oeis.org

1, 3, 10, 19, 42, 76, 156, 283, 570, 1046, 2096, 3890, 7784, 14582, 29168, 55051, 110106, 209062, 418128, 797806, 1595616, 3056868, 6113740, 11752674, 23505352, 45316896, 90633796, 175175816, 350351636, 678639146, 1357278296
Offset: 0

Views

Author

Clark Kimberling

Keywords

Showing 1-10 of 15 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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