A035287 -id:A035287 - cp's OEIS frontend

A165568 a(n) = -1 - 2n + n^2 + 2n^3 + n^4.

-1, 1, 31, 137, 391, 889, 1751, 3121, 5167, 8081, 12079, 17401, 24311, 33097, 44071, 57569, 73951, 93601, 116927, 144361, 176359, 213401, 255991, 304657, 359951, 422449, 492751, 571481, 659287, 756841, 864839, 984001, 1115071, 1258817, 1416031, 1587529, 1774151

Offset: 0

Paul Curtz, Sep 22 2009

Consider the Lyman spectrum of Hydrogen A005563(n)/A000290(n+1) = n*(n+2)/(n+1)^2 = 0/1, 3/4, 8/9, 15/16, ...
The first differences of these fractions are 3/4, 5/36, 7/144, 9/400, 11/900, 13/1764, 15/3136, ... = (2n+1)/(n*(n+1))^2.
Adding numerator and denominator of these first differences yields 1 + 2n + n^2 + 2n^3 + n^4 = A165563(n) = 3+4, 5+36, 7+144, ... = 1 + 2n + n^2*(n+1)^2 = A144396(n) + A035287(n+1) = A005408(n) + A035287(n+1).
Subtracting numerator from denominator, on the other hand, yields this sequence here: a(n) = A035287(n+1) - A005408(n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

[-1 -2*n +n^2 +2*n^3 +n^4: n in [0..40]]; // Vincenzo Librandi, May 21 2011

a(-1-n) = A165563(n). A165563(-1-n) = a(n).
a(n) = A165563(n) - 2 - 4*n = A165563(n) - A016825(n).
a(n) + A165563 + a(n) = 2*n^2*(1+n)^2 = 2*A035287(n+1).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 24.
G.f.: (-1 + 6*x + 16*x^2 + 2*x^3 + x^4)/(1-x)^5.

Edited and extended by R. J. Mathar, Feb 02 2010

A171733 a(2n)=A165568(n). a(2n+1)=A165563(n).

Original entry on oeis.org

-1, 1, 1, 7, 31, 41, 137, 151, 391, 409, 889, 911, 1751, 1777, 3121, 3151, 5167, 5201, 8081, 8119, 12079, 12121, 17401, 17447, 24311, 24361, 33097, 33151, 44071, 44129, 57569, 57631, 73951, 74017, 93601, 93671, 116927, 117001, 144361, 144439, 176359, 176441, 213401, 213487, 255991

Offset: 0

Paul Curtz, Dec 17 2009

Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A035287.

A171733(n)=if( bittest(n,0), n, -1-n) +(n\=2)^2 +2*n^3 +n^4

a(n) = n^3/8 +n^2/16 -15/32 +n^4/16 +(-1)^n*( n^3/8+3*n^2/16-n-17/32 ).
G.f.: ( 1-2*x-4*x^2+2*x^3-18*x^4+2*x^5-4*x^6-2*x^7+x^8 ) / ( (1+x)^4*(-1+x)^5 ).
a(n)= +a(n-1) +4*a(n-2) -4*a(n-3) -6*a(n-4) +6*a(n-5) +4*a(n-6) -4*a(n-7) -a(n-8) +a(n-9).

A219605 Square array T(n,k), read by antidiagonals: T(n,2k) = T(n,2k-1)n, T(n,2k+1) = T(n,2*k)+n, T(n,0) = 1.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 0, 2, 3, 1, 0, 3, 6, 4, 1, 0, 3, 8, 12, 5, 1, 0, 4, 16, 15, 20, 6, 1, 0, 4, 18, 45, 24, 30, 7, 1, 0, 5, 36, 48, 96, 35, 42, 8, 1, 0, 5, 38, 144, 100, 175, 48, 56, 9, 1, 0, 6, 76, 147, 400, 180, 288, 63, 72, 10, 1, 0, 6, 78, 441, 404, 900, 294, 441

Offset: 0

Philippe Deléham, Apr 12 2013

			Square array begins:
1..1....0....0....0....0....0....0.....0.....0...
1..2....2....3....3....4....4....5.....5.....5...
1..3....6....8...16...18...36...38....76....78...
1..4...12...15...45...48..144..147...441...444...
1..5...20...24...96..100..400..404..1616..1620...
1..6...30...35..175..180..900..905..4525..4530...
1..7...42...48..288..294.1764.1770.10620.10626...
1..8...56...63..441..448.3136.3143.22001.22008...
1..9...72...80..640..648.5184.5192.41536.41544...
1.10...90...99..891..900.8100.8109.72971.72980...
...

t[n_, k_] /; n < 0 || k < 0 = 0; t[n_, 0] = 1; t[n_, 1] = n+1; t[0, k_ /; k > 1] = 0; t[n_?Positive, k_?EvenQ] := t[n, k] = t[n, k-1]*n; t[n_?Positive, k_?OddQ] := t[n, k] = t[n, k-1] + n; Table[t[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Apr 19 2013 *)

T(n,0) = A000012(n).
T(n,1) = A000027(n).
T(n,2) = A002378(n+1).
T(n,3) = A005563(n).
T(n,4) = A152618(n+1).
T(n,5) = A045991(n+1).
T(n,6) = A035287(n+1).
T(0,k) = A019590(k+1).
T(1,k) = A008619(k+1).
T(2,k) = A123208(k).

A353435 Array read by descending antidiagonals: T(n,m) is the number of sequences of length n >= 0 with elements in 0..m-1 such that the Hankel matrix of any odd number of consecutive terms is invertible over the ring of integers modulo m >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 0, 1, 1, 4, 4, 4, 0, 1, 1, 2, 16, 0, 4, 0, 1, 1, 6, 4, 48, 0, 0, 0, 1, 1, 4, 36, 0, 144, 0, 0, 0, 1, 1, 6, 16, 180, 0, 320, 0, 0, 0, 1, 1, 4, 36, 0, 900, 0, 720, 0, 0, 0, 1, 1, 10, 16, 108, 0, 3744, 0, 1312, 0, 0, 0, 1

Offset: 0

Pontus von Brömssen, Apr 21 2022

T(n,m) is divisible by T(2,m) = A127473(n) for n >= 2, because if r and s are coprime to m, the sequence (x_1, ..., x_n) satisfies the conditions if and only if the sequence (r*s^0*x_1 mod m, ..., r*s^(n-1)*x_n mod m) does.

			Array begins:
  n\m| 1  2  3  4    5  6        7  8   9 10
  ---+--------------------------------------
   0 | 1  1  1  1    1  1        1  1   1  1
   1 | 1  1  2  2    4  2        6  4   6  4
   2 | 1  1  4  4   16  4       36 16  36 16
   3 | 1  0  4  0   48  0      180  0 108  0
   4 | 1  0  4  0  144  0      900  0 324  0
   5 | 1  0  0  0  320  0     3744  0   0  0
   6 | 1  0  0  0  720  0    15552  0   0  0
   7 | 1  0  0  0 1312  0    54216  0   0  0
   8 | 1  0  0  0 2400  0   189468  0   0  0
   9 | 1  0  0  0 3232  0   550728  0   0  0
  10 | 1  0  0  0 4560  0  1604088  0   0  0
  11 | 1  0  0  0 4656  0  3895560  0   0  0
  12 | 1  0  0  0 4928  0  9467856  0   0  0
  13 | 1  0  0  0 4368  0 19185516  0   0  0

Cf. A035287, A045991, A350364, A353433, A353436.
Rows: A000012 (n=0), A000010 (n=1), A127473 (n=2).
Columns: A000012 (m=1), A130716 (m=2), A166926 (m=4 and m=6).

For fixed n, T(n,m) is multiplicative with T(n,p^e) = T(n,p)*p^(n*(e-1)).
T(n,m) = A353436(n,m) if m is prime.
T(3,m) = (m-1)^2*(m-2) = A045991(m-1) if m is prime.
T(4,m) = (m-1)^2*(m-2)^2 = A035287(m-1) if m is prime.
Empirically: T(5,m) = (m-1)^2*(m-3)*(m^2-4*m+5) if m >= 3 is prime.
T(n,2) = 0 for n >= 3.
T(n,3) = 0 for n >= 5.
T(n,5) = 0 for n >= 23.

A374668 a(n) is the permanent of the n-th order Hankel matrix M whose generic element is given by M(i,j) = A317614(i+j-1) with i,j = 1, ..., n.

Original entry on oeis.org

1, 1, 31, 10254, 12238276, 41596930860, 309346186680924, 4522151204857137264, 116038936382978521700928, 4918677318766771695942334272, 323424014903141386787887115413440, 31725978444319999354629697685162941056, 4460612938377274751881312432310360618154240

Offset: 0

Stefano Spezia, Jul 15 2024

nonn

The Hankel transform of A317614 has the following polynomial as g.f. 1 + x - x^2 - 624*x^3 - 9216*x^4 + 138240*x^5 - 331776*x^6: the matrices are singular for n > 6.

			a(7) = 4522151204857137264:
  [  1,   4,  15,  32,  65, 108,  175]
  [  4,  15,  32,  65, 108, 175,  256]
  [ 15,  32,  65, 108, 175, 256,  369]
  [ 32,  65, 108, 175, 256, 369,  500]
  [ 65, 108, 175, 256, 369, 500,  671]
  [108, 175, 256, 369, 500, 671,  864]
  [175, 256, 369, 500, 671, 864, 1105]
which is the singular matrix M of minimal order.

N. J. A. Sloane, Transforms.

Cf. A317614.

Cf. A000583 (trace of M), A006010 (sum of the 1st row or column of M), A035287 (super- or subdiagonal sum of M), A346174, A374708 (k-th super- or subdiagonal sum of M).

A317614[n_]:=(1/2)*(n^3 + n*Mod[n,2]); a[n_]:=Permanent[Table[A317614[i+j-1], {i, n}, {j, n}]]; Join[{1}, Array[a, 12]]

A374708 Triangle T read by rows: T(n,k) = (n - k)n(4n^2 - 4nk + 2k^2 - 1 + (-1)^k)/4, with 0 <= k < n.

Original entry on oeis.org

1, 16, 4, 81, 36, 15, 256, 144, 80, 32, 625, 400, 255, 140, 65, 1296, 900, 624, 396, 240, 108, 2401, 1764, 1295, 896, 609, 364, 175, 4096, 3136, 2400, 1760, 1280, 864, 544, 256, 6561, 5184, 4095, 3132, 2385, 1728, 1215, 756, 369, 10000, 8100, 6560, 5180, 4080, 3100, 2320, 1620, 1040, 500

Offset: 1

Stefano Spezia, Jul 17 2024

T(n, k) is the k-th super- and subdiagonal sum of the Hankel matrix M(n) whose permanent is A374668(n).

			n\k|    0    1    2    3    4    5
---+------------------------------
1  |    1
2  |   16    4
3  |   81   36   15
4  |  256  144   80   32
5  |  625  400  255  140   65
6  | 1296  900  624  396  240  108
      ...
For n = 3 the matrix M is
  [ 1,  4, 15]
  [ 4, 15, 32]
  [15, 32, 65]
and therefore T(3, 0) = 1 + 15 + 65 = 81, T(3, 1) = 4 + 32 = 36, and T(3, 2) = 15.

Cf. A317614 (diagonal), A374668.
Cf. A333119, A330613.
Cf. A000583 (k=0), A035287 (k=1), A123865, A374709 (row sums).

T[n_,k_]:=(n-k)*n*(4*n^2 - 4*n*k+2*k^2-1+(-1)^k)/4; Table[T[n,k],{n,10},{k,0,n-1}]//Flatten

O.g.f.: x*(1 - 4*x^8*y^5 + x*(11 + 2*y) - x^7*y^4*(7 + 16*y) - x^2*(-11 + 6*y - 6*y^2) - x^5*y^2*(2 - 46*y - 3*y^2) - x^6*y^3*(-2 - 27*y + 4*y^2) - x^3*(-1 + 18*y + 38*y^2 - 2*y^3) - x^4*y*(2 + 14*y + 2*y^2 - y^3))/((1 - x)^5*(1 - x*y)^4*(1 + x*y)^2).

T(n,2) = A123865(n-1) for n > 1.

A163279 a(n) = (n^6 + 2n^5 + 2n^4 + n^3 + 2n)/2.

Original entry on oeis.org

4, 86, 705, 3364, 11630, 32514, 78211, 168200, 331704, 610510, 1062149, 1763436, 2814370, 4342394, 6507015, 9504784, 13574636, 19003590, 26132809, 35364020, 47166294, 62083186, 80740235, 103852824, 132234400, 166805054, 208600461

Offset: 1

Omar E. Pol, Nov 07 2009

Before this sequence, a(5) = 11630 was an uninteresting number, see Links section. - Omar E. Pol, Apr 25 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
Charles R Greathouse IV, Uninteresting numbers
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000290, A002378, A035287.

for n=1:354 a(n) = n^2*((n*(n+1))^2 + n*(n+1) + 2/n)/2; end
% Kyle Stern, Jan 05 2010

Array[Function[n, (n^6 + 2 n^5 + 2 n^4 + n^3 + 2 n)/2], {27}] (* or *)
Rest@ CoefficientList[Series[x (4 + 58 x + 187 x^2 + 95 x^3 + 17 x^4 - x^5)/(1 - x)^7, {x, 0, 27}], x] (* Michael De Vlieger, Apr 25 2016 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{4,86,705,3364,11630,32514,78211},30] (* Harvey P. Dale, Mar 08 2018 *)

a(n)=(n^6+n^3)/2+n^5+n^4+n \\ Charles R Greathouse IV, Jul 29 2011

G.f.: x*(4 + 58*x + 187*x^2 + 95*x^3 + 17*x^4 - x^5)/(1 - x)^7. - Ilya Gutkovskiy, Apr 25 2016

More terms from Kyle Stern, Jan 05 2010

A248720 a(n) = (n*(n+1))^5.

Original entry on oeis.org

0, 32, 7776, 248832, 3200000, 24300000, 130691232, 550731776, 1934917632, 5904900000, 16105100000, 40074642432, 92389579776, 199690286432, 408410100000, 796262400000, 1488827973632, 2682916351776, 4678757435232, 7923516800000, 13069123200000

Offset: 0

Eugene Chong, Oct 16 2014

This is the sequence (2^5)*A059860(n)= (2*binomial(n+1,2))^5, n >= 0. - Wolfdieter Lang, Nov 03 2014

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A059860, A002378 (n*(n+1)), A035287(n+1) ((n*(n+1))^2), A060459 ((n*(n+1))^3), A248619 ((n*(n+1))^4).

Magma
```
[(n*(n+1))^5: n in [0..30]];
```
Maple
```
[ seq(n^5*(n+1)^5, n = 0..100) ];
```

Table[(n (n + 1))^5, {n, 0, 70}] (* or *) CoefficientList[Series[32 x (x^8 + 232 x^7 + 5158 x^6 + 27664 x^5 + 47290 x^4 + 27664 x^3 + 5158 x^2 + 232 x + 1)/(1 - x)^11, {x, 0, 30}], x]
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{0,32,7776,248832,3200000,24300000,130691232,550731776,1934917632,5904900000,16105100000},20] (* Harvey P. Dale, Apr 23 2017 *)

a(n) = A002378(n)^5.
a(n) = 32*A059860(n) for n>0.
G.f.: 32*x*(x^8 + 232*x^7 + 5158*x^6 + 27664*x^5 + 47290*x^4 + 27664*x^3 + 5158*x^2 + 232*x + 1) / (1 - x)^11 (from A059860).
Sum_{n>=1} 1/a(n) = 126 - 35*Pi^2/3 - Pi^4/9. - Vaclav Kotesovec, Sep 25 2019
a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11). - Wesley Ivan Hurt, Jan 20 2024

Terms a(32) and beyond corrected by Andrew Howroyd, Feb 20 2018

A303987 Triangle read by rows: T(n, k) = (binomial(n,k)*binomial(n+k,k))^2 = A063007(n, k)^2, for n >= 0, k = 0..n.

Original entry on oeis.org

1, 1, 4, 1, 36, 36, 1, 144, 900, 400, 1, 400, 8100, 19600, 4900, 1, 900, 44100, 313600, 396900, 63504, 1, 1764, 176400, 2822400, 9922500, 7683984, 853776, 1, 3136, 571536, 17640000, 133402500, 276623424, 144288144, 11778624, 1, 5184, 1587600, 85377600, 1200622500, 5194373184, 7070119056, 2650190400, 165636900

Offset: 0

Wolfdieter Lang, May 14 2018

The row sums of this triangle are b(n) = A005259(n), for n >= 0. This sequence b was used in R. Apéry's 1979 proof of the irrationality of Zeta(3). See A005259 for references and links.

Row polynomials R(n, x) := Sum_{k=0..n} T(n, k)*x^k = hypergeometric([-n, -n, n+1, n+1], [1, 1, 1], x), hence b(n) = hypergeometric([-n, -n, n+1, n+1], [1, 1, 1], 1) (see the formula in A005259 given by K. A. Penson. This is the solution to Exercise 2.14 of the Koepf reference given there, p. 29).

			The triangle T begins:
n\k  0    1       2        3          4          5          6          7 ...
0:   1
1:   1    4
2:   1   36      36
3:   1  144     900      400
4:   1  400    8100    19600       4900
5:   1  900   44100   313600     396900      63504
6:   1 1764  176400  2822400    9922500    7683984     853776
7:   1 3136  571536 17640000  133402500  276623424  144288144   11778624
----------------------------------------------------------------------------
row n = 8:   1 5184 1587600 85377600 1200622500 5194373184 7070119056 2650190400 165636900,
row n = 9: 1 8100 3920400 341510400 8116208100 63631071504 176752976400 169612185600 47869064100 2363904400,
row n = 10: 1 12100 8820900 1177862400 44188244100 572679643536 2828047622400 5446435737600 3877394192100 853369488400 34134779536.
...

Cf. A000984, A002894, A005259, A008459, A063007.

The column sequences (without zeros) are A000012, A035287(n+1) = 4*A000217(n)^2, 36*A288876, 400*A000579(n+6)^2, 4900*A000581(n+8)^2, 63504*A001287(n+10)^2, ...

Flat(List([0..10],n->List([0..n],k->(Binomial(n,k)*Binomial(n+k,k))^2))); # Muniru A Asiru, May 15 2018

T[n_, k_] := (Gamma[k + n + 1]/(Gamma[k + 1]^2*Gamma[-k + n + 1]))^2;
Flatten[Table[T[n, k], {n, 0, 8}, {k, 0, n}]] (* Peter Luschny, May 14 2018 *)

T(n, k) = (binomial(n,k)*binomial(n+k,k))^2 = A063007(n, k)^2, for n >= 0 and k = 0..n.
T(n, k) = (binomial(n+k, 2*k)*cbi(k))^2, with cbi(k) = A000984(k) = binomial(2*k, k), and cbi(k)^2 = A002894(k).
G.f. for column sequences (without leading zeros):
  cbi(k)^2*P2(2*k, x)/(1 - x)^(4*k+1), with the row polynomials of A008459 (Pascal entries squared) P2(2*k, x) = Sum_{j=0..2*k} A008459(2*k, j)*x^j. For a proof see the general comment in A288876 on the diagonals and columns of A008459.

A174426 Denominator of fractions in A171676.

Original entry on oeis.org

4, 36, 36, 144, 144, 144, 400, 400, 400, 400, 900, 900, 900, 900, 900, 1764, 1764, 1764, 1764, 1764, 1764, 3136, 3136, 3136, 3136, 3136, 3136, 3136, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 5184, 8100, 8100, 8100, 8100, 8100, 8100, 8100, 8100, 8100

Offset: 0

Paul Curtz, Nov 28 2010

(Antidiagonal writing) fractions are 3/4, 5/36,5/36, 7/144,7/144,7/144, . Numerator:A171676. Rydberg-Ritz.

n times A035287(n+2).

cp's OEIS Frontend

A165568 a(n) = -1 - 2*n + n^2 + 2*n^3 + n^4.

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A171733 a(2n)=A165568(n). a(2n+1)=A165563(n).

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A219605 Square array T(n,k), read by antidiagonals: T(n,2*k) = T(n,2*k-1)*n, T(n,2*k+1) = T(n,2*k)+n, T(n,0) = 1.

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A353435 Array read by descending antidiagonals: T(n,m) is the number of sequences of length n >= 0 with elements in 0..m-1 such that the Hankel matrix of any odd number of consecutive terms is invertible over the ring of integers modulo m >= 1.

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A374668 a(n) is the permanent of the n-th order Hankel matrix M whose generic element is given by M(i,j) = A317614(i+j-1) with i,j = 1, ..., n.

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A374708 Triangle T read by rows: T(n,k) = (n - k)*n*(4*n^2 - 4*n*k + 2*k^2 - 1 + (-1)^k)/4, with 0 <= k < n.

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A163279 a(n) = (n^6 + 2n^5 + 2n^4 + n^3 + 2n)/2.

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A248720 a(n) = (n*(n+1))^5.

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A165568 a(n) = -1 - 2n + n^2 + 2n^3 + n^4.

A219605 Square array T(n,k), read by antidiagonals: T(n,2k) = T(n,2k-1)n, T(n,2k+1) = T(n,2*k)+n, T(n,0) = 1.

A374708 Triangle T read by rows: T(n,k) = (n - k)n(4n^2 - 4nk + 2k^2 - 1 + (-1)^k)/4, with 0 <= k < n.