A084902 -id:A084902 - cp's OEIS frontend

A084901 a(n) = 4^(n-2)n(5*n+3)/2.

0, 1, 13, 108, 736, 4480, 25344, 136192, 704512, 3538944, 17367040, 83623936, 396361728, 1853882368, 8573157376, 39258685440, 178241142784, 803158884352, 3594887626752, 15994458210304, 70781061038080, 311711546474496

Offset: 0

Paul Barry, Jun 10 2003

Binomial transform of A084900. Third binomial transform of heptagonal numbers A000566. Fourth binomial transform of (0,1,5,0,0,0,...).

Coefficients in the hypergeometric series identity 1 - 13*x/(x + 12) + 108*x*(x - 1)/((x + 12)*(x + 16)) - 736*x*(x - 1)*(x - 2)/((x + 12)*(x + 16)*(x + 20)) + ... = 0, valid in the half-plane Re(x) > 0. Cf. A276289 and A077616. - Peter Bala, May 30 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-48,64).

Cf. A084902, A077616, A276289.

List([0..30], n-> 2^(2*n-5)*n*(5*n+3)); # G. C. Greubel, Jun 06 2019

[2^(2*n-5)*n*(5*n+3): n in [0..30]]; // G. C. Greubel, Jun 06 2019

Table[2^(2*n-5)*n*(5*n+3), {n,0,30}] (* G. C. Greubel, Jun 06 2019 *)
LinearRecurrence[{12,-48,64},{0,1,13},30] (* or *) CoefficientList[ Series[-((x (1+x))/(-1+4 x)^3),{x,0,30}],x] (* Harvey P. Dale, Jul 14 2021 *)

vector(30, n, n--; 2^(2*n-5)*n*(5*n+3)) \\ G. C. Greubel, Jun 06 2019

[2^(2*n-5)*n*(5*n+3) for n in (0..30)] # G. C. Greubel, Jun 06 2019

G.f.: x*(1+x)/(1-4*x)^3.
E.g.f.: x*(2 + 5*x)*exp(4*x)/2. - G. C. Greubel, Jun 06 2019
a(n) = 12*a(n-1)-48*a(n-2)+64*a(n-3). - Wesley Ivan Hurt, May 28 2021

A218016 Triangle, read by rows, where T(n,k) = k!C(n, k)5^(n-k) for n>=0, k=0..n.

Original entry on oeis.org

1, 5, 1, 25, 10, 2, 125, 75, 30, 6, 625, 500, 300, 120, 24, 3125, 3125, 2500, 1500, 600, 120, 15625, 18750, 18750, 15000, 9000, 3600, 720, 78125, 109375, 131250, 131250, 105000, 63000, 25200, 5040, 390625, 625000, 875000, 1050000, 1050000, 840000, 504000, 201600, 40320

Offset: 0

Vincenzo Librandi, Nov 10 2012

Triangle formed by the derivatives of x^n evaluated at x=5.
Sum(T(n,k), k=0..n) = A080954(n) (see the Formula section of A080954). . Also:
first column:    A000351;
second column:   A053464;
third column:  2*A084902;
fourth column: 6*A081143.

			Triangle begins:
1;
5,      1;
25,     10,     2;
125,    75,     30,     6;
625,    500,    300,    120,     24;
3125,   3125,   2500,   1500,    600,     120;
15625,  18750,  18750,  15000,   9000,    3600,   720;
78125,  109375, 131250, 131250,  105000,  63000,  25200,  5040;
390625, 625000, 875000, 1050000, 1050000, 840000, 504000, 201600, 40320; etc.

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A000351, A053464, A080954, A081143, A084902, A090802, A217629, A218017.

[Factorial(n)/Factorial(n-k)*5^(n-k): k in [0..n], n in [0..10]];

Flatten[Table[n!/(n-k)!*5^(n-k), {n, 0, 10}, {k, 0, n}]]

T(n,k) = 5^(n-k)*n!/(n-k)! for n>=0, k=0..n.

E.g.f. (by columns): exp(5x)*x^k.

A116156 a(n) = 5^n * n*(n + 1).

Original entry on oeis.org

0, 10, 150, 1500, 12500, 93750, 656250, 4375000, 28125000, 175781250, 1074218750, 6445312500, 38085937500, 222167968750, 1281738281250, 7324218750000, 41503906250000, 233459472656250, 1304626464843750, 7247924804687500

Offset: 0

Mohammad K. Azarian, Apr 08 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (15,-75,125).

Cf. A007758, A036289, A084902, A128796.

List([0..30], n-> 5^n*n*(n+1)); # G. C. Greubel, May 10 2019

[(n^2+n)*5^n: n in [0..30]]; // Vincenzo Librandi, Feb 28 2013

I:=[0,10,150]; [n le 3 select I[n] else 15*Self(n-1)-75*Self(n-2)+125*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Feb 28 2013

Table[(n^2 + n) 5^n, {n, 0, 30}] (* or *) CoefficientList[Series[10 x/(1 - 5 x)^3, {x, 0, 30}], x](* Vincenzo Librandi, Feb 28 2013 *)

a(n)=(n^2+n)*5^n \\ Charles R Greathouse IV, Feb 28 2013

[5^n*n*(n+1) for n in (0..30)] # G. C. Greubel, May 10 2019

G.f.: 10*x/(1-5*x)^3. - Vincenzo Librandi, Feb 28 2013
a(n) = 15*a(n-1) -75*a(n-2) +125*a(n-3). - Vincenzo Librandi, Feb 28 2013
a(n) = 10*A084902(n). - Bruno Berselli, Feb 28 2013
E.g.f.: 5*x*(2 + 5*x)*exp(5*x). - G. C. Greubel, May 10 2019
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = 1 - 4*log(5/4).
Sum_{n>=1} (-1)^(n+1)/a(n) = 6*log(6/5) - 1. (End)

A173113 a(n) = binomial(n + 10, 10) * 5^n.

Original entry on oeis.org

1, 55, 1650, 35750, 625625, 9384375, 125125000, 1519375000, 17092968750, 180425781250, 1804257812500, 17222460937500, 157872558593750, 1396564941406250, 11970556640625000, 99754638671875000

Offset: 0

Zerinvary Lajos, Feb 10 2010

With a different offset, number of n-permutations (n>=10) of 6 objects: t, u, v, z, x, y with repetition allowed, containing exactly ten (10) u's.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (55,-1375,20625,-206250,1443750,-7218750,25781250,-64453125,107421875,-107421875,48828125).

Cf. A053464, A084902, A081143, A036071, A050982, A140405, A140220, A140346, A140520.

[5^n*Binomial(n+10, 10): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

Table[Binomial[n + 10, 10]*5^n, {n, 0, 20}]

a(n) = C(n + 10, 10)*5^n, n>=0.
G.f.: 1/(1-5*x)^11. - Vincenzo Librandi, Oct 15 2011
From Amiram Eldar, Sep 01 2022: (Start)
Sum_{n>=0} 1/a(n) = 184261655/63 - 13107200*log(5/4).
Sum_{n>=0} (-1)^n/a(n) = 503884800*log(6/5) - 11575501585/126. (End)

cp's OEIS Frontend

A084901 a(n) = 4^(n-2)*n*(5*n+3)/2.

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A218016 Triangle, read by rows, where T(n,k) = k!*C(n, k)*5^(n-k) for n>=0, k=0..n.

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A116156 a(n) = 5^n * n*(n + 1).

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GAP

Magma

Magma

Mathematica

PARI

Sage

Formula

A173113 a(n) = binomial(n + 10, 10) * 5^n.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A084901 a(n) = 4^(n-2)n(5*n+3)/2.

A218016 Triangle, read by rows, where T(n,k) = k!C(n, k)5^(n-k) for n>=0, k=0..n.