A272126 -id:A272126 - cp's OEIS frontend

A272127 a(n) = 1680n^4 - 168n^2 + 128*n + 1.

1, 1641, 26465, 134953, 427905, 1046441, 2172001, 4026345, 6871553, 11010025, 16784481, 24577961, 34813825, 47955753, 64507745, 85014121, 110059521, 140268905, 176307553, 218881065, 268735361, 326656681, 393471585, 470046953, 557289985, 656148201, 767609441

Offset: 0

Vincenzo Librandi, Apr 25 2016

This is the polynomial Qbar(4,n) in Brent. See A160485 for the triangle of coefficients (with signs) of the Qbar polynomials. - Peter Bala, Jan 21 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Richard P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014. (page 16).
Richard P. Brent, Generalising Tuenter's binomial sums, Journal of Integer Sequences, 18 (2015), Article 15.3.2.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A014641, A272126, A160485, A245244, A272129.

[1680*n^4-168*n^2+128*n+1: n in [0..50]];

Table[1680 n^4 - 168 n^2 + 128 n + 1, {n, 0, 30}]
LinearRecurrence[{5,-10,10,-5,1},{1,1641,26465,134953,427905},30] (* Harvey P. Dale, Nov 27 2017 *)

a(n) = 1680*n^4 - 168*n^2 + 128*n + 1; \\ Altug Alkan, Apr 30 2016

O.g.f.: (1+1636*x+18270*x^2+19028*x^3+1385*x^4)/(1-x)^5.
E.g.f.: (1+1640*x+11592*x^2+10080*x^3+1680*x^4)*exp(x).
a(n) = (2*n+1)*(840*n^3-420*n^2+126*n+1).
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5), for n>4
See page 7 in Brent's paper:
a(n) = (2*n+1)^2*A272126(n) - 2*n*(2*n+1)*A272126(n-1).
A272128(n) = (2*n+1)^2*a(n) - 2*n*(2*n+1)*a(n-1).
From Peter Bala, Jan 22 2019: (Start)
a(n) = 1/4^n * Sum_{k = 0..n} (2*k + 1)^8 * binomial(2*n + 1, n - k).
a(n-1) = 2/4^n * binomial(2*n,n) * ( 1 + 3^8*(n - 1)/(n + 1) + 5^8*(n - 1)*(n - 2)/((n + 1)*(n + 2)) + 7^8*(n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)) + ... ). (End)

A272129 a(n) = 32n^2 - 56n + 25.

Original entry on oeis.org

25, 1, 41, 145, 313, 545, 841, 1201, 1625, 2113, 2665, 3281, 3961, 4705, 5513, 6385, 7321, 8321, 9385, 10513, 11705, 12961, 14281, 15665, 17113, 18625, 20201, 21841, 23545, 25313, 27145, 29041, 31001, 33025, 35113, 37265, 39481, 41761, 44105, 46513, 48985

Offset: 0

Vincenzo Librandi, Apr 26 2016

Subsequence of A001844.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Richard P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014 (page 16).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001844, A272126, A272127, A272128, A272131, A272132, A272133, A245244.

Magma
```
[32*n^2 - 56*n + 25: n in [0..50]];
```

[32*n^2-56*n+25$n=0..40]; # Muniru A Asiru, Jan 28 2019

Table[32 n^2 - 56 n + 25, {n, 0, 40}]
LinearRecurrence[{3,-3,1},{25,1,41},50] (* Harvey P. Dale, Jul 03 2018 *)

lista(nn) = for(n=0, nn, print1(32*n^2-56*n+25, ", ")); \\ Altug Alkan, Apr 26 2016

O.g.f.: (25 - 74*x + 113*x^2)/(1-x)^3.
E.g.f.: (25 - 24*x + 32*x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
n*a(n) = 1 + 3^5*(n-1)/(n+1) + 5^5*((n-1)*(n-2))/((n+1)*(n+2)) + ... for n >= 1. See A245244. - Peter Bala, Jan 19 2019

A272128 a(n) = 30240n^5-25200n^4+5040n^3+7320n^2-2638*n+1.

Original entry on oeis.org

1, 14763, 628805, 5501167, 24943689, 79549811, 203823373, 449807415, 890712977, 1624547899, 2777745621, 4508793983, 7011864025, 10520438787, 15310942109, 21706367431, 30079906593, 40858578635, 54526858597, 71630306319, 92779195241, 118652141203

Offset: 0

Vincenzo Librandi, Apr 25 2016

This is the polynomial Qbar(5,n) in Brent. See A160485 for the triangle of coefficients (with signs) of the Qbar polynomials. - Peter Bala, Feb 01 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Richard P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014. (page 16).
Richard P. Brent, Generalising Tuenter's binomial sums, Journal of Integer Sequences, 18 (2015), Article 15.3.2.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A014641, A272126, A272127.

[30240*n^5-25200*n^4+5040*n^3+7320*n^2-2638*n+1: n in [0..40]];

Table[30240 n^5 - 25200 n^4 + 5040 n^3 + 7320 n^2 - 2638 n + 1, {n, 0, 30}]

a(n)=30240*n^5-25200*n^4+5040*n^3+7320*n^2-2638*n+1 \\ Charles R Greathouse IV, Apr 30 2016

O.g.f.: (1 + 14757*x + 540242*x^2 + 1949762*x^3 + 1073517*x^4 + 50521*x^5)/(1-x)^6.
E.g.f.: (1 + 14762*x + 299640*x^2 + 609840*x^3 + 277200*x^4 + 30240*x^5)*exp(x).
a(n) = (2*n+1)*(15120*n^4-20160*n^3+12600*n^2-2640*n+1).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
See page 7 in Brent's paper: a(n) = (2*n+1)^2*A272127(n) - 2*n*(2*n+1)*A272127(n-1).
From Peter Bala, Feb 01 2019: (Start)
a(n) = 1/4^n * Sum_{k = 0..n} (2*k + 1)^10 * binomial(2*n + 1, n - k).
a(n-1) = 2/4^n * binomial(2*n,n) * ( 1 + 3^10*(n - 1)/(n + 1) + 5^10*(n - 1)*(n - 2)/((n + 1)*(n + 2)) + 7^10*(n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)) + ... ). (End)

cp's OEIS Frontend

A272127 a(n) = 1680n^4 - 168n^2 + 128*n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A272129 a(n) = 32n^2 - 56n + 25.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A272128 a(n) = 30240n^5-25200n^4+5040n^3+7320n^2-2638*n+1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

cp's OEIS Frontend

A272127 a(n) = 1680*n^4 - 168*n^2 + 128*n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A272129 a(n) = 32*n^2 - 56*n + 25.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A272128 a(n) = 30240*n^5-25200*n^4+5040*n^3+7320*n^2-2638*n+1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A272127 a(n) = 1680n^4 - 168n^2 + 128*n + 1.

A272129 a(n) = 32n^2 - 56n + 25.

A272128 a(n) = 30240n^5-25200n^4+5040n^3+7320n^2-2638*n+1.