A258807 -id:A258807 - cp's OEIS frontend

A300656 Triangle read by rows: T(n,k) = 30k^2(n-k)^2 + 1; n >= 0, 0 <= k <= n.

1, 1, 1, 1, 31, 1, 1, 121, 121, 1, 1, 271, 481, 271, 1, 1, 481, 1081, 1081, 481, 1, 1, 751, 1921, 2431, 1921, 751, 1, 1, 1081, 3001, 4321, 4321, 3001, 1081, 1, 1, 1471, 4321, 6751, 7681, 6751, 4321, 1471, 1, 1, 1921, 5881, 9721, 12001, 12001, 9721, 5881, 1921, 1

Offset: 0

Kolosov Petro, Mar 10 2018

From Kolosov Petro, Apr 12 2020: (Start)
Let A(m, r) = A302971(m, r) / A304042(m, r).
Let L(m, n, k) = Sum_{r=0..m} A(m, r) * k^r * (n - k)^r.
Then T(n, k) = L(2, n, k).
Fifth power can be expressed as row sum of triangle T(n, k).
T(n, k) is symmetric: T(n, k) = T(n, n-k). (End)

			Triangle begins:
--------------------------------------------------------------------------
k=    0     1     2      3      4      5      6      7     8     9    10
--------------------------------------------------------------------------
n=0:  1;
n=1:  1,    1;
n=2:  1,   31,    1;
n=3:  1,  121,  121,     1;
n=4:  1,  271,  481,   271,     1;
n=5:  1,  481, 1081,  1081,   481,     1;
n=6:  1,  751, 1921,  2431,  1921,   751,     1;
n=7:  1, 1081, 3001,  4321,  4321,  3001,  1081,     1;
n=8:  1, 1471, 4321,  6751,  7681,  6751,  4321,  1471,    1;
n=9:  1, 1921, 5881,  9721, 12001, 12001,  9721,  5881, 1921,    1;
n=10: 1, 2431, 7681, 13231, 17281, 18751, 17281, 13231, 7681, 2431,   1;

Kolosov Petro, Rows n = 0..2078 of triangle, flattened.
Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
Petro Kolosov, Polynomial identity involving binomial theorem and Faulhaber's formula, 2023.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.

Various cases of L(m, n, k): A287326(m=1), This sequence (m=2), A300785(m=3). See comments for L(m, n, k).
Row sums give the nonzero terms of A002561.
Cf. A000584, A287326, A007318, A077028, A294317, A068236, A302971, A304042, A002561, A258807, A158558, A094053, A024003, A316349.

T:=Flat(List([0..9],n->List([0..n],k->30*k^2*(n-k)^2+1))); # Muniru A Asiru, Oct 24 2018

[[30*k^2*(n-k)^2+1: k in [0..n]]: n in [0..12]]; // G. C. Greubel, Dec 14 2018

a:=(n,k)->30*k^2*(n-k)^2+1: seq(seq(a(n,k),k=0..n),n=0..9); # Muniru A Asiru, Oct 24 2018

T[n_, k_] := 30 k^2 (n - k)^2 + 1; Column[
Table[T[n, k], {n, 0, 10}, {k, 0, n}], Center] (* Kolosov Petro, Apr 12 2020 *)
f[n_]:=Table[SeriesCoefficient[(1 + 26 y + 336 y^2 + 326 y^3 + 31 y^4 + x^2 (1 + 116 y + 486 y^2 + 116 y^3 + y^4) + x (-2 - 82 y - 882 y^2 - 502 y^3 + 28 y^4))/((-1 + x)^3 (-1 + y)^5), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n}]; Flatten[Array[f, 11, 0]] (* Stefano Spezia, Oct 30 2018 *)

t(n, k) = 30*k^2*(n-k)^2+1
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
/* Print initial 9 rows of triangle as follows */ trianglerows(9)

[[30*k^2*(n-k)^2+1 for k in range(n+1)] for n in range(12)] # G. C. Greubel, Dec 14 2018

From Kolosov Petro, Apr 12 2020: (Start)
T(n, k) = 30 * k^2 * (n-k)^2 + 1.
T(n, k) = 30 * A094053(n,k)^2 + 1.
T(n, k) = A158558((n-k) * k).
T(n+2, k) = 3*T(n+1, k) - 3*T(n, k) + T(n-1, k), for n >= k.
Sum_{k=1..n} T(n, k) = A000584(n).
Sum_{k=0..n-1} T(n, k) = A000584(n).
Sum_{k=0..n} T(n, k) = A002561(n).
Sum_{k=1..n-1} T(n, k) = A258807(n).
Sum_{k=1..n-1} T(n, k) = -A024003(n), n > 1.
Sum_{k=1..r} T(n, k) = A316349(2,r,0)*n^0 - A316349(2,r,1)*n^1 + A316349(2,r,2)*n^2. (End)
G.f.: (1 + 26*y + 336*y^2 + 326*y^3 + 31*y^4 + x^2*(1 + 116*y + 486*y^2 + 116*y^3 + y^4) + x*(-2 - 82*y - 882*y^2 - 502*y^3 + 28*y^4))/((-1 + x)^3*(-1 + y)^5). - Stefano Spezia, Oct 30 2018

A319214 a(n) = phi(n^5 - 1)/5 where phi is A000010.

Original entry on oeis.org

6, 22, 120, 280, 1240, 1120, 5400, 5280, 12960, 12880, 45240, 24752, 90240, 59160, 96000, 141984, 346880, 163800, 540000, 326720, 588984, 585120, 1523280, 582400, 1728000, 1203840, 2294712, 1758096, 4692408, 1388000, 6480000, 3787200, 5416800, 4783680, 7440000

Offset: 2

Seiichi Manyama, Sep 13 2018

Seiichi Manyama, Table of n, a(n) for n = 2..1000
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

Row 5 of A369291.
Cf. A000010, A258807 (n^5-1).
phi(n^b - 1)/b: A319210 (b=2), A319213 (b=3), this sequence (b=5).

Table[EulerPhi[n^5-1]/5,{n,2,40}] (* Harvey P. Dale, Feb 09 2019 *)

PARI
```
{a(n) = eulerphi(n^5-1)/5}
```

Sum_{k=1..n} a(k) = c * n^6 + O((n*log(n))^5), where c = (1/30) * Product_{p prime == 1 (mod 5)} (1 - 5/p^2) * Product_{p prime !== 1 (mod 5)} (1 - 1/p^2) = 0.019389107739... . - Amiram Eldar, Dec 09 2024

A258808 a(n) = n^7 - 1.

Original entry on oeis.org

0, 127, 2186, 16383, 78124, 279935, 823542, 2097151, 4782968, 9999999, 19487170, 35831807, 62748516, 105413503, 170859374, 268435455, 410338672, 612220031, 893871738, 1279999999, 1801088540, 2494357887, 3404825446, 4586471423, 6103515624, 8031810175

Offset: 1

Vincenzo Librandi, Jun 11 2015

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Subsequence of A181126.
Cf. A258806.
Cf. similar sequences listed in A258807.

Magma
```
[n^7-1: n in [1..40]];
```

I:=[0,127,2186,16383, 78124,279935,823542,2097151]; [n le 8 select I[n] else 8*Self(n-1) - 28*Self(n-2)+56*Self(n-3)-70*Self(n-4)+56*Self(n-5) - 28*Self(n-6) +8*Self(n-7)-Self(n-8): n in [1..40]];

Table[n^7 - 1, {n, 1, 40}] (* or *) LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 127, 2186, 16383, 78124, 279935, 823542, 2097151}, 40]

[n^7-1 for n in (1..40)] # Bruno Berselli, Jun 11 2015

G.f.: x^2*(127 + 1170*x + 2451*x^2 + 1156*x^3 + 141*x^4 - 6*x^5 + x^6)/(1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
a(n) = -A024005(n). [Bruno Berselli, Jun 11 2015]
a(n) = (n-1)*A053716(n). - Michel Marcus, Aug 21 2015

A258809 a(n) = n^8 - 1.

Original entry on oeis.org

0, 255, 6560, 65535, 390624, 1679615, 5764800, 16777215, 43046720, 99999999, 214358880, 429981695, 815730720, 1475789055, 2562890624, 4294967295, 6975757440, 11019960575, 16983563040, 25599999999, 37822859360, 54875873535, 78310985280, 110075314175

Offset: 1

Vincenzo Librandi, Jun 11 2015

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A024006, A060890.

Cf. similar sequences listed in A258807.

Magma
```
[n^8-1: n in [1..40]];
```

Table[n^8 - 1, {n, 33}] (* or *) LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {0, 255, 6560, 65535, 390624, 1679615, 5764800, 16777215, 43046720}, 40]

G.f.: x^2*(225 + 4535*x + 14595*x^2 + 18069*x^3 + 569*x^4 + 3999*x^5 - 2511*x^6 + 1079*x^7 - 270*x^8 + 30*x^9) / (1 - x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9).
a(n) = (n - 1)*(n + 1)*(n^2 + 1)*(n^4 + 1) = -A024006(n). [Bruno Berselli, Jun 12 2015]

A258810 a(n) = n^9 - 1.

Original entry on oeis.org

0, 511, 19682, 262143, 1953124, 10077695, 40353606, 134217727, 387420488, 999999999, 2357947690, 5159780351, 10604499372, 20661046783, 38443359374, 68719476735, 118587876496, 198359290367, 322687697778, 511999999999, 794280046580, 1207269217791, 1801152661462

Offset: 1

Vincenzo Librandi, Jun 11 2015

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A024007, similar sequences listed in A258807.

Magma
```
[n^9-1: n in [1..40]];
```

I:=[0,511,19682, 262143,1953124,10077695,40353606,134217727,387420488, 999999999]; [n le 10 select I[n] else 10*Self(n-1)-45*Self(n-2)+120*Self(n-3)-210*Self(n-4)+252*Self(n-5)-210*Self(n-6)+120*Self(n-7)-45*Self(n-8)+10*Self(n-9)-Self(n-10): n in [1..40]];

Table[n^9 - 1, {n, 33}] (* or *) LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 511, 19682, 262143, 1953124, 10077695, 40353606, 134217727, 387420488, 999999999}, 40]

G.f.: x^2*(511 + 14572*x + 88318*x^2 + 156064*x^3 + 88360*x^4 + 14524*x^5 + 538*x^6 - 8*x^7 + x^8)/(1-x)^10.

a(n) = 10*a(n-1)-45*a(n-2)+120*a(n-3)-210*a(n-4)+252*a(n-5)-210*a(n-6)+120*a(n-7)-45*a(n-8)+10*a(n-9)-a(n-10).

A258812 a(n) = n^11 - 1.

Original entry on oeis.org

0, 2047, 177146, 4194303, 48828124, 362797055, 1977326742, 8589934591, 31381059608, 99999999999, 285311670610, 743008370687, 1792160394036, 4049565169663, 8649755859374, 17592186044415, 34271896307632, 64268410079231, 116490258898218, 204799999999999

Offset: 1

Vincenzo Librandi, Jun 11 2015

Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A024009.

Cf. similar sequences listed in A258807.

Magma
```
[n^11-1: n in [1..40]];
```

Table[n^11 - 1, {n, 33}] (* or *) LinearRecurrence[{12, -66, 220, -495, 792, -924, 792, -495, 220, -66, 12, -1}, {0, 2047, 177146, 4194303, 48828124, 362797055, 1977326742, 8589934591, 31381059608, 99999999999, 285311670610, 743008370687}, 40]

G.f.: x^2* (2047 + 152582*x + 2203653*x^2 + 9737784*x^3 + 15724710*x^4 + 9737652*x^5 + 2203818*x^6 + 152472*x^7 + 2091*x^8 - 10*x^9 + x^10) / (1 - x)^12.
a(n) = 12*a(n-1) - 66*a(n-2) + 220*a(n-3) - 495*a(n-4) + 792*a(n-5) - 924*a(n-6) + 792*a(n-7) - 495*a(n-8) + 220*a(n-9) - 66*a(n-10) + 12*a(n-11) - a(n-12).
a(n) = -A024009(n). [Bruno Berselli, Jun 12 2015]

cp's OEIS Frontend

A300656 Triangle read by rows: T(n,k) = 30*k^2*(n-k)^2 + 1; n >= 0, 0 <= k <= n.

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A319214 a(n) = phi(n^5 - 1)/5 where phi is A000010.

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Mathematica

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A258808 a(n) = n^7 - 1.

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Magma

Magma

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A258809 a(n) = n^8 - 1.

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Magma

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A258810 a(n) = n^9 - 1.

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Magma

Magma

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A258812 a(n) = n^11 - 1.

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Magma

Mathematica

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A300656 Triangle read by rows: T(n,k) = 30k^2(n-k)^2 + 1; n >= 0, 0 <= k <= n.