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.

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A015507 a(1) = 1, a(n) = Sum_{k=1..n-1} ((6^k - 1)/5)*a(k).

Original entry on oeis.org

1, 1, 8, 352, 91520, 142405120, 1328924579840, 74403829376081920, 24994031979330942894080, 50376471215620688640734003200, 609214555257707874214915513922355200, 44204249911340791820804231319883906967142400
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Sequences with the recurrence a(n) = (m^(n-1) + m-2)*a(n-1)/(m-1): A036442 (m=2), A015502 (m=3), A015503 (m=4), A015506 (m=5), this sequence (m=6), A015508 (m=7), A015509 (m=8), A015511 (m=9), A015512 (m=10), A015513 (m=11), A015515 (m=12).

Programs

  • Magma
    [n le 2 select 1 else ((6^(n-1)+4)/5)*Self(n-1): n in [1..15]]; // Vincenzo Librandi, Nov 12 2012
  • Mathematica
    a[n_,m_]:= a[n,m]= If[n<3, 1, (m^(n-1)+m-2)*a[n-1,m]/(m-1)];
Table[a[n,6], {n,20}] (* G. C. Greubel, Apr 29 2023 *)
  • SageMath
    @CachedFunction
def A015507(n): return 1 if (n<3) else (6^(n-1)+4)*A015507(n-1)/5
[A015507(n) for n in range(1,21)] # G. C. Greubel, Apr 29 2023

Formula

a(n) = ((6^(n-1) + 4)/5) * a(n-1). - Vincenzo Librandi, Nov 12 2012

A015508 a(1) = 1, a(n) = Sum_{k=1..n-1} ((7^k - 1)/6)*a(k).

Original entry on oeis.org

1, 1, 9, 522, 209322, 586520244, 11501075464596, 1578614616119517768, 1516734501782248791012168, 10200952598655696033329019125136, 480252779391204632593567857157274897424, 158269444415262012661462389451687149577571916192
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Sequences with the recurrence a(n) = (m^(n-1) + m-2)*a(n-1)/(m-1): A036442 (m=2), A015502 (m=3), A015503 (m=4), A015506 (m=5), A015507 (m=6), this sequence (m=7), A015509 (m=8), A015511 (m=9), A015512 (m=10), A015513 (m=11), A015515 (m=12).

Programs

  • Magma
    [n le 2 select 1 else ((7^(n-1) + 5)/6)*Self(n-1): n in [1..15]]; // Vincenzo Librandi, Nov 12 2012
  • Mathematica
    a[n_, m_]:= a[n, m]= If[n<3, 1, (m^(n-1) + m-2)*a[n-1,m]/(m-1)];
Table[a[n,7], {n,30}] (* G. C. Greubel, Apr 30 2023 *)
  • SageMath
    @CachedFunction # a = A015508
def a(n,m): return 1 if (n<3) else (m^(n-1)+m-2)*a(n-1,m)/(m-1)
[a(n,7) for n in range(1,31)] # G. C. Greubel, Apr 30 2023

Formula

a(n) = ((7^(n-1) + 5)/6) * a(n-1). - Vincenzo Librandi, Nov 12 2012

A015509 a(1) = 1, a(n) = Sum_{k=1..n-1} ((8^k - 1)/7)*a(k).

Original entry on oeis.org

1, 1, 10, 740, 433640, 2030302480, 76034827876000, 22779578222682344000, 54596862986901017252624000, 1046838176230046602563156976288000, 160576277008444677145920980328106246720000
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Sequences with the recurrence a(n) = (m^(n-1) + m-2)*a(n-1)/(m-1): A036442 (m=2), A015502 (m=3), A015503 (m=4), A015506 (m=5), A015507 (m=6), A015508 (m=7), this sequence (m=8), A015511 (m=9), A015512 (m=10), A015513 (m=11), A015515 (m=12).

Programs

  • Magma
    [n le 2 select 1 else ((8^(n-1)+6)/7)*Self(n-1): n in [1..15]]; // Vincenzo Librandi, Nov 12 2012
  • Mathematica
    a[n_, m_]:= a[n, m]= If[n<3, 1, (m^(n-1) +m-2)*a[n-1,m]/(m-1)];
Table[a[n,8], {n,30}] (* G. C. Greubel, Apr 30 2023 *)
  • SageMath
    @CachedFunction # a = A015509
def a(n,m): return 1 if (n<3) else (m^(n-1)+m-2)*a(n-1,m)/(m-1)
[a(n,8) for n in range(1,31)] # G. C. Greubel, Apr 30 2023

Formula

a(n) = ((8^(n-1) + 6)/7) * a(n-1). - Vincenzo Librandi, Nov 12 2012

A015511 a(1) = 1, a(n) = Sum_{k=1..n-1} ((9^k - 1)/8)*a(k).

Original entry on oeis.org

1, 1, 11, 1012, 830852, 6133349464, 407444538242984, 243599680968409330048, 1310771150941736627904810368, 63477451180042308935531134194562816, 27666523379269090447091129488519658150671616
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Sequences with the recurrence a(n) = (m^(n-1) + m-2)*a(n-1)/(m-1): A036442 (m=2), A015502 (m=3), A015503 (m=4), A015506 (m=5), A015507 (m=6), A015508 (m=7), A015509 (m=8), this sequence (m=9), A015512 (m=10), A015513 (m=11), A015515 (m=12).

Programs

  • Magma
    [n le 2 select 1 else ((9^(n-1)+7)/8)*Self(n-1): n in [1..15]]; // Vincenzo Librandi, Nov 12 2012
  • Mathematica
    a[n_, m_]:= a[n, m]= If[n<3, 1, (m^(n-1) +m-2)*a[n-1,m]/(m-1)];
Table[a[n,9], {n,30}] (* G. C. Greubel, May 03 2023 *)
Join[{1}, Table[7^n*QPochhammer[-1/7, 9, n]/2^(3*n + 1), {n, 2, 12}]] (* Vaclav Kotesovec, May 03 2023 *)
  • SageMath
    @CachedFunction # a = A015511
def a(n, m): return 1 if (n<3) else (m^(n-1)+m-2)*a(n-1, m)/(m-1)
[a(n,9) for n in range(1, 31)] # G. C. Greubel, May 03 2023

Formula

a(n) = ((9^(n-1) + 7)/8) * a(n-1). - Vincenzo Librandi, Nov 12 2012
a(n) ~ QPochhammer(-63, 1/9) * 3^(n*(n-1)) / 2^(3*n+7). - Vaclav Kotesovec, May 03 2023

A015512 a(1) = 1, a(n) = Sum_{k=1..n-1} ((10^k - 1)/9)*a(k).

Original entry on oeis.org

1, 1, 12, 1344, 1494528, 16607195136, 1845258665951232, 2050289046842405289984, 22780991231839211526404702208, 2531221268231904597902043824359735296, 2812468078063201791652852780757078172764209152
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Sequences with the recurrence a(n) = (m^(n-1) + m-2)*a(n-1)/(m-1): A036442 (m=2), A015502 (m=3), A015503 (m=4), A015506 (m=5), A015507 (m=6), A015508 (m=7), A015509 (m=8), A015511 (m=9), this sequence (m=10), A015513 (m=11), A015515 (m=12).

Programs

  • Magma
    [n le 2 select 1 else ((10^(n-1) + 8)/9)*Self(n-1): n in [1..15]]; // Vincenzo Librandi, Nov 12 2012
  • Mathematica
    a[n_, m_]:= a[n, m]= If[n<3, 1, (m^(n-1) +m-2)*a[n-1,m]/(m-1)];
Table[a[n, 10], {n, 30}] (* G. C. Greubel, May 03 2023 *)
  • SageMath
    @CachedFunction # a = A015512
def a(n, m): return 1 if (n<3) else (m^(n-1)+m-2)*a(n-1, m)/(m-1)
[a(n,10) for n in range(1, 31)] # G. C. Greubel, May 03 2023

Formula

a(n) = ((10^(n-1) + 8)/9) * a(n-1). - Vincenzo Librandi, Nov 12 2012

A015513 a(1) = 1, a(n) = Sum_{k=1..n-1} ((11^k - 1)/10)*a(k).

Original entry on oeis.org

1, 1, 13, 1742, 2552030, 41102995180, 7281683317103260, 14189947350338830620680, 304174136317707285574697584520, 71722670512982436329410134761448960400, 186030135925835196854820049614502274473787544400
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Sequences with the recurrence a(n) = (m^(n-1) + m-2)*a(n-1)/(m-1): A036442 (m=2), A015502 (m=3), A015503 (m=4), A015506 (m=5), A015507 (m=6), A015508 (m=7), A015509 (m=8), A015511 (m=9), A015512 (m=10), this sequence (m=11), A015515 (m=12).

Programs

  • Magma
    [n le 2 select 1 else ((11^(n-1) + 9)/10) * Self(n-1): n in [1..15]]; // Vincenzo Librandi, Nov 12 2012
  • Mathematica
    a[n_, m_]:= a[n, m]= If[n<3, 1, (m^(n-1) +m-2)*a[n-1,m]/(m-1)];
Table[a[n, 10], {n, 30}] (* G. C. Greubel, May 03 2023 *)
  • SageMath
    def a(n, m) -> int: # a = A015513
    return 1 if (n<3) else (m^(n-1)+m-2)*a(n-1, m)//(m-1)
[a(n, 11) for n in range(1,31)] # G. C. Greubel, May 03 2023

Formula

a(n) = ((11^(n-1) + 9)/10) * a(n-1). - Vincenzo Librandi, Nov 12 2012

A015515 a(1) = 1, a(n) = Sum_{k=1..n-1} ((12^k - 1)/11)*a(k).

Original entry on oeis.org

1, 1, 14, 2212, 4171832, 94375183504, 25618521062894816, 83450744014073963641408, 3262026661649164626974053098368, 1530121919008888925087797696409496422656, 8612828743790947623482719127044813092555596516864
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Sequences with the recurrence a(n) = (m^(n-1) + m-2)*a(n-1)/(m-1): A036442 (m=2), A015502 (m=3), A015503 (m=4), A015506 (m=5), A015507 (m=6), A015508 (m=7), A015509 (m=8), A015511 (m=9), A015512 (m=10), A015513 (m=11), this sequence (m=12).

Programs

  • Magma
    [n le 2 select 1 else ((12^(n-1) + 10)/11) * Self(n-1): n in [1..15]]; // Vincenzo Librandi, Nov 12 2012
  • Mathematica
    Join[{1},RecurrenceTable[{a[2]==1,a[n]==(12^(n-1)+10)/11 a[n-1]},a,{n,12}]] (* Harvey P. Dale, Mar 10 2013 *)
  • SageMath
    def a(n, m) -> int: # a = A015515
    return 1 if (n<3) else (m^(n-1)+m-2)*a(n-1, m)//(m-1)
[a(n, 12) for n in range(1,31)] # G. C. Greubel, May 03 2023

Formula

a(n) = ((12^(n-1) + 10)/11) * a(n-1). - Vincenzo Librandi, Nov 12 2012

A109980 Number of Delannoy paths of length n with no (1,1)-steps on the line y=x.

Original entry on oeis.org

1, 2, 8, 36, 172, 852, 4324, 22332, 116876, 618084, 3296308, 17702412, 95627580, 519170004, 2830862532, 15494401116, 85091200620, 468692890308, 2588521289812, 14330490031020, 79509491551772, 442019710668852
Offset: 0

Views

Author

Emeric Deutsch, Jul 06 2005

Keywords

Comments

A Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1).
Equals left border of triangle A152250 and INVERTi transform of A001850, the Delannoy numbers: (1, 3, 13, 63, 321, ...). - Gary W. Adamson, Nov 30 2008
Hankel transform is A036442. First column of Riordan array ((1-x)/(1+x), x/(1+3x+2x^2))^{-1}. - Paul Barry, Apr 27 2009

Examples

			a(2)=8 because we have NDE, EDN, NENE, NEEN, ENNE, ENEN, NNEE and EENN.

Links

Crossrefs

First column of A109979.
Cf. A152250.

Programs

  • Maple
    g:=1/(z+sqrt(1-6*z+z^2)): gser:=series(g,z=0,28): 1,seq(coeff(gser,z^n),n=1..25);
  • Mathematica
    CoefficientList[Series[1/(x+Sqrt[1-6*x+x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)

Formula

G.f.: 1/(z + sqrt(1 - 6*z + z^2)).
Moment representation: a(n) = 0^n/3 + (1/Pi)*Integral_{x=3-2*sqrt(2)..3+2*sqrt(2)} x^n*sqrt(-x^2+6x-1)/(x*(6-x)) dx. - Paul Barry, Apr 27 2009
From Gary W. Adamson, Aug 23 2011: (Start)
a(n) is the upper left term in M^n, M = an infinite square production matrix as follows:
2, 2, 0, 0, 0, 0, ...
2, 1, 2, 0, 0, 0, ...
2, 1, 1, 2, 0, 0, ...
2, 1, 1, 1, 2, 0, ...
2, 1, 1, 1, 1, 2, ...
... (End)
D-finite with recurrence: n*a(n) = 3*(4*n-3)*a(n-1) - (37*n-57)*a(n-2) + 6*(n-3)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 2^(1/4) * (1 + sqrt(2))^(2*n+3) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 18 2012, simplified Dec 24 2017

A127945 Hankel transform of central coefficients of (1+k*x-2x^2)^n, k arbitrary integer.

Original entry on oeis.org

1, -4, -32, 512, 16384, -1048576, -134217728, 34359738368, 17592186044416, -18014398509481984, -36893488147419103232, 151115727451828646838272, 1237940039285380274899124224, -20282409603651670423947251286016
Offset: 0

Views

Author

Paul Barry, Feb 08 2007

Keywords

Comments

Hankel transform of A098332. The Hankel transform of e.g.f. Bessel_I(0,2*sqrt(-2)x) and its k-th binomial transforms, are given by a(n). In general, the Hankel transform of e.g.f. Bessel_I(0,2*sqrt(m)x) and its binomial transforms is 2^n*m^C(n+1,2).
Unsigned version is A036442. - Philippe Deléham, Dec 11 2008

Links

Crossrefs

Cf. A036442, A098332.

Programs

  • Magma
    [2^n*(-2)^Binomial(n+1,2): n in [0..25]]; // G. C. Greubel, May 01 2018
  • Mathematica
    Table[2^n*(-2)^Binomial[n+1,2], {n, 0, 25}] (* G. C. Greubel, May 01 2018 *)
  • PARI
    for(n=0,25, print1(2^n*(-2)^binomial(n+1,2), ", ")) \\ G. C. Greubel, May 01 2018

Formula

a(n) = (cos(Pi*n/2) - sin(Pi*n/2))*4^n*2^C(n,2).
a(n) = 2^n*(-2)^C(n+1,2).

A178185 Numerator of Sum_{k=1..n} 1/2^((k^2 + 3*k)/2).

Original entry on oeis.org

1, 9, 145, 4641, 297025, 38019201, 9732915457, 4983252713985, 5102850779120641, 10450638395639072769, 42805814868537642061825, 350665235403060363770470401, 5745299216843741000015387049985
Offset: 1

Views

Author

Artur Jasinski, May 21 2010

Keywords

Comments

Series of the kind m^((k^2 + 3*k)/2) from k=1 to n were studied by Bernoulli and Euler.

Links

Crossrefs

Cf. A178184-A178193.
Cf. A036442 (denominators).

Programs

  • Mathematica
    aa = {}; m = 1/2; sum = 0; Do[sum = sum + m^((n + 3) n/2); AppendTo[aa, Numerator[sum]], {n, 1, 20}]; aa
Numerator[Table[Sum[1/2^((k^2 + 3*k)/2), {k, 1, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Apr 10 2024 *)
  • PARI
    a(n) = numerator(sum(k=1, n, (1/2)^((k^2+3*k)/2))); \\ Michel Marcus, Sep 09 2013

Formula

a(n) = 2^(n+1)*a(n-1) + 1, a(1) = 1. - Alexandre Herrera, Mar 23 2024
a(n) ~ c * A036442(n+1) = c * 2^(n*(n+3)/2), where c = 2^(1/8) * EllipticTheta[2, 0, 1/Sqrt[2]] - 3 [in Mathematica] = 2^(1/8) * JacobiTheta2(0, 1/sqrt(2)) - 3 [in Maple] = 0.2832651213103077325876855404508588684521230759134794956... - Vaclav Kotesovec, Apr 10 2024
Previous Showing 11-20 of 28 results. Next

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