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 1-8 of 8 results.

A016161 Expansion of g.f. 1/((1-5*x)*(1-7*x)).

Original entry on oeis.org

1, 12, 109, 888, 6841, 51012, 372709, 2687088, 19200241, 136354812, 964249309, 6798573288, 47834153641, 336059778612, 2358521965909, 16540171339488, 115933787267041, 812299450322412, 5689910849522509
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Also, this is the number of incongruent integer-edged Heron triangles whose circumdiameter is the product of n distinct primes each of shape 4k + 1. Cf. A003462, A109021. - R. K. Guy, Jan 31 2007

Links

Crossrefs

Cf. A003462, A080962, A081200, A109021.
Cf. A016162, A016163, A016164, A016165, A016166.

Programs

  • Magma
    [n le 2 select 12^(n-1) else 12*Self(n-1) -35*Self(n-2): n in [1..30]]; // G. C. Greubel, Nov 09 2024
  • Mathematica
    CoefficientList[Series[1/((1-5x)(1-7x)),{x,0,30}],x] (* or *) LinearRecurrence[ {12,-35},{1,12},30] (* Harvey P. Dale, Nov 16 2021 *)
  • PARI
    Vec(1/((1-5*x)*(1-7*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012
  • SageMath
    A016161=BinaryRecurrenceSequence(12,-35,1,12)
[A016161(n) for n in range(31)] # G. C. Greubel, Nov 09 2024

Formula

From R. J. Mathar, Sep 18 2008: (Start)
a(n) = (7^(n+1) - 5^(n+1))/2 = A081200(n+1).
Binomial transform of A080962. (End)
a(n) = 7*a(n-1) + 5^n. - Vincenzo Librandi, Feb 09 2011
E.g.f.: exp(5*x)*(7*exp(2*x) - 5)/2. - Stefano Spezia, Oct 25 2023

A075500 Stirling2 triangle with scaled diagonals (powers of 5).

Original entry on oeis.org

1, 5, 1, 25, 15, 1, 125, 175, 30, 1, 625, 1875, 625, 50, 1, 3125, 19375, 11250, 1625, 75, 1, 15625, 196875, 188125, 43750, 3500, 105, 1, 78125, 1984375, 3018750, 1063125, 131250, 6650, 140, 1, 390625, 19921875
Offset: 1

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(5*z) - 1)*x/5) - 1.

Examples

			[1]; [5,1]; [25,15,1]; ...; p(3,x) = x(25 + 15*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*     1
*     5       1
*    25      15       1
*   125     175      30       1
*   625    1875     625      50      1
*  3125   19375   11250    1625     75    1
* 15625  196875  188125   43750   3500  105   1
* 78125 1984375 3018750 1063125 131250 6650 140 1
(End)

Links

Crossrefs

Columns 1-7 are A000351, A016164, A075911-A075915. Row sums are A005011(n-1).
Cf. A075499, A075501.

Programs

  • Maple
    # The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> 5^n, 9); # Peter Luschny, Jan 28 2016
  • Mathematica
    Flatten[Table[5^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[5^#&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
  • PARI
    for(n=1, 11, for(m=1, n, print1(5^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n, m) = (5^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*5)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 5m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-5k*x), m >= 1.
E.g.f. for m-th column: (((exp(5x)-1)/5)^m)/m!, m >= 1.

A075911 Third column of triangle A075500.

Original entry on oeis.org

1, 30, 625, 11250, 188125, 3018750, 47265625, 728906250, 11133203125, 168996093750, 2554931640625, 38523925781250, 579858642578125, 8717878417968750, 130968170166015625, 1966522521972656250, 29517837677001953125, 442967564392089843750, 6646513462066650390625
Offset: 0

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

The e.g.f. given below is Sum_{m=0..2} A075513(3,m)*exp(5*(m+1)*x)/2!.

Links

Crossrefs

Cf. A016164, A075912.

Programs

  • Mathematica
    LinearRecurrence[{30,-275,750},{1,30,625},30] (* Harvey P. Dale, Oct 06 2017 *)
  • PARI
    Vec(1/((1-5*x)*(1-10*x)*(1-15*x)) + O(x^30)) \\ Colin Barker, Dec 11 2015

Formula

a(n) = A075500(n+3, 3) = (5^n)*S2(n+3, 3) with S2(n, m) = A008277(n, m) (Stirling2).
a(n) = (5^n - 8*10^n + 9*15^n)/2.
G.f.: 1/Product_{k=1..3} (1 - 5*k*x).
E.g.f.: (d^3/dx^3)(((exp(5*x)-1)/5)^3)/3! = (exp(5*x) - 8*exp(10*x) + 9*exp(15*x))/2!.
a(n) = 30*a(n-1) - 275*a(n-2) + 750*a(n-3) for n > 2. - Colin Barker, Dec 11 2015

A075913 Fifth column of triangle A075500.

Original entry on oeis.org

1, 75, 3500, 131250, 4344375, 132890625, 3855156250, 107765625000, 2933008203125, 78271552734375, 2058270703125000, 53524929199218750, 1380066321044921875, 35349237725830078125, 900813505310058593750, 22863955398559570312500, 578500758117828369140625
Offset: 0

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

The e.g.f. given below is Sum_{m=0..4}(A075513(5,m)*exp(5*(m+1)*x))/4!.

Links

Crossrefs

Cf. A000351, A016164, A075911, A075912, A075914.

Programs

  • Mathematica
    Table[5^n*(1 - 2^(n+6) + 2*3^(n+5) - 4^(n+5) + 5^(n+4))/24, {n, 0, 20}] (* Vaclav Kotesovec, Dec 12 2015 *)
  • PARI
    Vec(1/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)) + O(x^30)) \\ Colin Barker, Dec 12 2015

Formula

a(n) = A075500(n+5, 5) = (5^n)*S2(n+5, 5) with S2(n, m) = A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..4}(A075513(5, m)*((m+1)*5)^n)/4!.
G.f.: 1/Product_{k=1..5}(1-5*k*x).
E.g.f.: (d^5/dx^5)((((exp(5*x)-1)/5)^5)/5!) = (exp(5*x) - 64*exp(10*x) + 486*exp(15*x) - 1024*exp(20*x) + 625*exp(25*x))/4!.
G.f.: 1 / ((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)). - Colin Barker, Dec 12 2015

A075914 Sixth column of triangle A075500.

Original entry on oeis.org

1, 105, 6650, 330750, 14266875, 560896875, 20682062500, 728227500000, 24779833203125, 821666548828125, 26708267167968750, 854772944238281250, 27023254648193359375, 846046877171630859375, 26282219820458984375000, 811330550012329101562500
Offset: 0

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

The e.g.f. given below is Sum_{m=0..5}(A075513(6,m)*exp(5*(m+1)*x))/5!.

Links

Crossrefs

Cf. A000351, A016164, A075911, A075912, A075913, A075915.

Programs

  • Mathematica
    Table[5^(n-1) * (-1 + 5*2^(5+n) + 5*2^(11+2*n) - 10*3^(5+n) - 5^(6+n) + 6^(5+n))/24, {n, 0, 20}] (* Vaclav Kotesovec, Dec 12 2015 *)
  • PARI
    Vec(1/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)*(1-30*x)) + O(x^30)) \\ Colin Barker, Dec 12 2015

Formula

a(n) = A075500(n+6, 6) = (5^n)*S2(n+6, 6) with S2(n, m) = A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..5}(A075513(6, m)*((m+1)*5)^n)/5!.
G.f.: 1/Product_{k=1..6}(1-5*k*x).
E.g.f.: (d^6/dx^6)((((exp(5*x)-1)/5)^6)/6!) = (-exp(5*x) + 160*exp(10*x) - 2430*exp(15*x) + 10240*exp(20*x) - 15625*exp(25*x) + 7776*exp(30*x))/5!.
G.f.: 1 / ((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)*(1-30*x)). - Colin Barker, Dec 12 2015

A075915 Seventh column of triangle A075500.

Original entry on oeis.org

1, 140, 11550, 735000, 39991875, 1960612500, 89303500000, 3853850000000, 159664583203125, 6409926960937500, 251055710800781250, 9641722822265625000, 364483553427490234375, 13602971247133789062500, 502386213470141601562500, 18394848021467285156250000
Offset: 0

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

The e.g.f. given below is Sum_{m=0..6}(A075513(7,m)exp(5*(m+1)*x))/6!.

Links

Crossrefs

Cf. A000351, A016164, A075911, A075912, A075913, A075914.

Programs

  • Mathematica
    Table[5^(n-1) * (1 - 3*2^(7 + n) - 5*2^(14 + 2*n) + 5*3^(7 + n) + 3*5^(7 + n) - 6^(7 + n) + 7^(6 + n))/144, {n, 0, 20}] (* Vaclav Kotesovec, Dec 12 2015 *)
  • PARI
    Vec(1/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)*(1-30*x)*(1-35*x)) + O(x^30)) \\ Colin Barker, Dec 12 2015

Formula

a(n) = A075500(n+7, 7) = (5^n)S2(n+7, 7) with S2(n, m) = A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..6}(A075513(7, m)*(5*(m+1))^n)/6!.
G.f.: 1/Product_{k=1..7}(1-5k*x).
E.g.f.: (d^7/dx^7)((((exp(5x)-1)/5)^7)/7!) = (exp(5*x) - 384*exp(10*x) + 10935*exp(15*x) - 81920*exp(20*x) + 234375*exp(25*x) - 279936*exp(30*x) + 117649*exp(35*x))/6!.
G.f.: 1 / ((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)*(1-30*x)*(1-35*x)). - Colin Barker, Dec 12 2015

A248340 a(n) = 10^n - 5^n.

Original entry on oeis.org

0, 5, 75, 875, 9375, 96875, 984375, 9921875, 99609375, 998046875, 9990234375, 99951171875, 999755859375, 9998779296875, 99993896484375, 999969482421875, 9999847412109375, 99999237060546875, 999996185302734375, 9999980926513671875
Offset: 0

Views

Author

Vincenzo Librandi, Oct 05 2014

Keywords

Links

Crossrefs

Cf. sequences of the form k^n-5^n: A005062 (k=6), A121213 (k=7), A191468 (k=8), A191466 (k=9), this sequence (k=10), A139743 (k=11).
Cf. A000225, A000351, A011557, A016164.

Programs

  • Magma
    [10^n-5^n: n in [0..30]];
  • Mathematica
    Table[10^n - 5^n, {n,0,30}]
CoefficientList[Series[5 x/((1-5 x)(1-10 x)), {x, 0, 30}], x]
  • Python
    def A248340(n): return pow(10,n) - pow(5,n)
print([A248340(n) for n in range(41)]) # G. C. Greubel, Nov 13 2024

Formula

G.f.: 5*x/((1-5*x)*(1-10*x)).
a(n) = 15*a(n-1) - 50*a(n-2).
a(n) = 5^n*(2^n-1) = A000351(n) * A000225(n) = A011557(n) - A000351(n).
a(n) = 5*A016164(n-1).
a(n) = A016164(n) - A011557(n).
E.g.f.: exp(10*x) - exp(5*x). - G. C. Greubel, Nov 13 2024

A102765 Array read by antidiagonals: T(n, k) = ((n+4)^k-(n-1)^k)/5.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 5, 13, 0, 1, 7, 25, 51, 0, 1, 9, 43, 125, 205, 0, 1, 11, 67, 259, 625, 819, 0, 1, 13, 97, 477, 1555, 3125, 3277, 0, 1, 15, 133, 803, 3355, 9331, 15625, 13107, 0, 1, 17, 175, 1261, 6505, 23517, 55987, 78125, 52429, 0, 1, 19, 223, 1875, 11605
Offset: 0

Views

Author

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 10 2005

Keywords

Comments

Consider a 5x5 matrix M =
[n, 1, 1, 1, 1]
[1, n, 1, 1, 1]
[1, 1, n, 1, 1]
[1, 1, 1, n, 1]
[1, 1, 1, 1, n].
The n-th row of the array contains the values of the non diagonal elements of M^k, k=0,1,.... (Corresponding diagonal entry = non diagonal entry + (n-1)^k.)
For row r we have polynomial ((r+4)^n-(r-1)^n)/5. Corresponding g.f.s: x/((1-(r-1)x)(1-(r+4)x))
If r(n) denotes a row sequence, r(n+1)/r(n) converges to n+4.
Triangle T(n, k) = (4^(n-k-1)-(-1)^(n-k-1))/5*(binomial(k+(n-k-1),n-k-1)) gives coefficients for polynomials for the columns of the array. First four polynomial are:
1
3 + 2*k
13 + 9*k + 3*k^2
51 + 52*k + 18*k^2 + 4*k^3
...

Examples

			Array begins:
  0, 1,  3, 13,  51,  205, ...
  0, 1,  5, 25, 125,  625, ...
  0, 1,  7, 43, 259, 1555, ...
  0, 1,  9, 67, 477, 3355, ...
  0, 1, 11, 97, 803, 6505, ...
  ...

Crossrefs

Cf. A015521 (for n=0), A000351 (for n=1), A003464 (for n=2), A016130 (for n=3), A016140 (for n=4), A016153 (for n=5), A016164 (for n=6), A016174 (for n=7), A016184 (for n=8), A015441 (for n=-1), A091005 (for n=-2).

Programs

  • PARI
    MM(n,N)=local(M);M=matrix(n,n);for(i=1,n, for(j=1,n,if(i==j,M[i,j]=N,M[i,j]=1)));M
for(k=0,10, for(i=0,10,print1((MM(5,k)^i)[1,2],","));print())
  • PARI
    p(n,k)=((n+4)^k-(n-1)^k)/5
for(k=0,10, for(i=0,10,print1(p(k,i),","));print())
  • PARI
    for(k=0,10, for(i=0,10,print1(polcoeff(x/((1-(k-1)*x)*(1-(k+4)*x)),i),","));print())
Showing 1-8 of 8 results.

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

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