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.

Previous Showing 11-15 of 15 results.

A051947 Partial sums of A034263.

Original entry on oeis.org

1, 10, 49, 168, 462, 1092, 2310, 4488, 8151, 14014, 23023, 36400, 55692, 82824, 120156, 170544, 237405, 324786, 437437, 580888, 761530, 986700, 1264770, 1605240, 2018835, 2517606, 3115035, 3826144, 4667608, 5657872, 6817272, 8168160, 9735033, 11544666
Offset: 0

Views

Author

Barry E. Williams, Dec 21 1999

Keywords

References

  • Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Cf. A034263.
Cf. A093561 ((4, 1) Pascal, column m=6).

Programs

Formula

a(n) = C(n+5, 5)*(2n+3)/3.
G.f.: (1+3*x)/(1-x)^7.
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=0} 1/a(n) = 2161/28 - 768*log(2)/7.
Sum_{n>=0} (-1)^n/a(n) = 192*Pi/7 - 624*log(2)/7 - 657/28. (End)

Extensions

Corrected by T. D. Noe, Nov 09 2006

A099856 Expansion of (1+3*x)/(1-3*x).

Original entry on oeis.org

1, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, 1062882, 3188646, 9565938, 28697814, 86093442, 258280326, 774840978, 2324522934, 6973568802, 20920706406, 62762119218, 188286357654, 564859072962, 1694577218886, 5083731656658, 15251194969974, 45753584909922
Offset: 0

Views

Author

Paul Barry, Oct 28 2004

Keywords

Comments

A099858 gives a Chebyshev transform. Binomial transform is A083420.
Hankel transform is 1, -18, 0, 0, 0, 0, 0, 0, 0, ... - Philippe Deléham, Dec 13 2011

Links

Crossrefs

Cf. A025192, A083420, A093561, A099858.

Programs

  • Mathematica
    CoefficientList[Series[(1+3x)/(1-3x),{x,0,30}],x] (* or *) Join[{1}, NestList[3#&,6,30]] (* Harvey P. Dale, Nov 08 2011 *)
  • PARI
    Vec((1+3*x)/(1-3*x) + O(x^40)) \\ Michel Marcus, Dec 11 2015

Formula

a(n) = 2*3^n - 0^n.
a(n) = A025192(n+1), n > 0. - R. J. Mathar, Sep 02 2008
a(n) = Sum_{k=0..n} A093561(n,k)*2^k. - Philippe Deléham, Dec 13 2011
From Elmo R. Oliveira, Aug 23 2024: (Start)
E.g.f.: 2*exp(3*x) - 1.
a(n) = 3*a(n-1) for n > 1. (End)

Extensions

a(26)-a(28) from Elmo R. Oliveira, Aug 23 2024

A052181 Partial sums of A050483.

Original entry on oeis.org

1, 12, 72, 300, 990, 2772, 6864, 15444, 32175, 62920, 116688, 206856, 352716, 581400, 930240, 1449624, 2206413, 3287988, 4807000, 6906900, 9768330, 13616460, 18729360, 25447500, 34184475, 45439056, 59808672, 78004432, 100867800, 129389040, 164727552, 208234224
Offset: 0

Views

Author

Barry E. Williams, Jan 26 2000

Keywords

References

  • Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Cf. A027819, A050483.
Cf. A093561 ((4, 1) Pascal, column m=8).

Programs

  • Maple
    a:=n->(sum((numbcomp(n,8)), j=7..n))/2:seq(a(n), n=8..31); # Zerinvary Lajos, Aug 26 2008
  • Mathematica
    Table[(n + 2)*Binomial[n + 7, 7]/2, {n, 0, 40}] (* Amiram Eldar, Feb 11 2022 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,12,72,300,990,2772,6864,15444,32175},40] (* Harvey P. Dale, Sep 06 2024 *)

Formula

a(n) = A027819(n+1)/7.
a(n) = (n+2)*C(n+7, 7)/2.
G.f.: (1+3*x)/(1-x)^9.
a(n) = C(n+2, 2)*C(n+7, 6)/7. - Zerinvary Lajos, Jul 29 2005
From Amiram Eldar, Feb 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 41783/300 - 14*Pi^2.
Sum_{n>=0} (-1)^n/a(n) = 7*Pi^2 - 2688*log(2)/5 + 91343/300. (End)

A055843 Expansion of (1+3*x)/(1-x)^10.

Original entry on oeis.org

1, 13, 85, 385, 1375, 4147, 11011, 26455, 58630, 121550, 238238, 445094, 797810, 1379210, 2309450, 3759074, 5965487, 9253475, 14060475, 20967375, 30735705, 44352165, 63081525, 88529025, 122713500, 168152556, 227961228, 305965660, 406833460, 536222500, 700950052
Offset: 0

Views

Author

Barry E. Williams, May 30 2000

Keywords

Comments

Partial sums of A052181.

References

  • Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Cf. A052181.
Cf. A093561 ((4, 1) Pascal, column m=9).

Programs

  • GAP
    List([0..30], n-> (4*n+9)*Binomial(n+8, 8)/9 ); # G. C. Greubel, Jan 21 2020
  • Magma
    [(4*n+9)*Binomial(n+8, 8)/9: n in [0..30]]; // G. C. Greubel, Jan 21 2020
  • Maple
    seq( (4*n+9)*binomial(n+8, 8)/9, n=0..30); # G. C. Greubel, Jan 21 2020
  • Mathematica
    Table[4*Binomial[n+9,9] - 3*Binomial[n+8,8], {n,0,30}] (* G. C. Greubel, Jan 21 2020 *)
  • PARI
    vector(31, n, (4*n+5)*binomial(n+7, 8)/9) \\ G. C. Greubel, Jan 21 2020
  • Sage
    [(4*n+9)*binomial(n+8, 8)/9 for n in (0..30)] # G. C. Greubel, Jan 21 2020

Formula

a(n) = (4*n+9)*binomial(n+8, 8)/9.
G.f.: (1+3*x)/(1-x)^10.
a(n) = 4*binomial(n+9,9) - 3*binomial(n+8,8). - G. C. Greubel, Jan 21 2020
Sum_{n>=0} 1/a(n) = 9437184*Pi/24035 + 56623104*log(2)/24035 - 482087736/168245. - Amiram Eldar, Feb 17 2023

A050483 Partial sums of A051947.

Original entry on oeis.org

1, 11, 60, 228, 690, 1782, 4092, 8580, 16731, 30745, 53768, 90168, 145860, 228684, 348840, 519384, 756789, 1081575, 1519012, 2099900, 2861430, 3848130, 5112900, 6718140, 8736975, 11254581, 14369616, 18195760, 22863368, 28521240, 35338512, 43506672, 53241705, 64786371
Offset: 0

Views

Author

Barry E. Williams, Dec 26 1999

Keywords

References

  • Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Cf. A051947.
Cf. A093561 ((4, 1) Pascal, column m=7).

Programs

  • Mathematica
    Table[(4*n + 7)*Binomial[n + 6, 6]/7, {n, 0, 40}] (* Amiram Eldar, Feb 15 2022 *)

Formula

a(n) = C(n+6, 6)*(4n+7)/7.
G.f.: (1+3*x)/(1-x)^8. - proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009
Sum_{n>=0} 1/a(n) = 57344*Pi/663 - 114688*log(2)/221 + 295372/3315. - Amiram Eldar, Feb 15 2022
Previous Showing 11-15 of 15 results.

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