A290053 -id:A290053 - cp's OEIS frontend

A290761 Irregular triangle, read by rows, of coefficients of polynomials that are the "nonstandard" factor of polynomials yielding the columns (up to sign) of triangle A290053, beginning with column 3.

3, 5, -6, 16, 1, 7, 16, 28, 0, 15, 225, 1265, 3707, 7120, 4900, -6480, 27648, 3, 83, 961, 6201, 24708, 60700, 87968, 85056, 0, 63, 2457, 41580, 404866, 2532971, 10651177, 30102338, 56577724, 72856616, 36562176, -51101568, 298598400, 9, 531, 14010, 219106

Offset: 1

Gregory Gerard Wojnar, Aug 09 2017

The polynomials come in pairs: first of odd degree; second of even degree 1 greater, whose constant term is always zero. Observations: All coefficients are positive except for the linear coefficients of the first polynomial in each pair, which are always negative. From the first of one pair to the first of the next pair, the degree always grows by 4. The "standard" factors of polynomials yielding the columns of triangle A290053 (beginning with column 3) are always of the form (1/A053657(k+2))*(N + k + 2) in odd rows of this triangle A290761, and of the form (N/A053657(k+2))*(N + k + 3)^2 in even rows of this triangle, where k is the row number. See examples.

			The first rows of the triangle are parsed as follows:
3, 5, -6, 16;
1, 7, 16, 28, 0;
15, 225, 1265, 3707, 7120, 4900, -6480, 27648;
3, 83, 961, 6201, 24708, 60700, 87968, 85056, 0;
63, 2457, 41580, 404866, 2532971, 10651177, 30102338, 56577724, 72856616, 36562176, -51101568, 298598400;
9, 531, 14010, 219106, 2266137, 16325259, 83797380, 307998768, 802828704, 1433652560, 1651979520, 1239918336, 0.
The associated full polynomials giving the columns of triangle A290053 are then:
(1/24) * (N + 3) * (3*N^3 + 5*N^2 - 6*N + 16);
(N/48) * (N + 5)^2 * (1*N^3 + 7*N^2 + 16*N + 28);
(1/5760) * (N + 5) * (15*N^7 + 225*N^6 + 1265*N^5 + 3707*N^4 + 7120*N^3 + 4900*N^2 - 6480*N + 27648);
(N/11520) * (N + 7)^2 * (3*N^7 + 83*N^6 + 961*N^5 + 6201*N^4 + 24708*N^3 + 60700*N^2 + 87968*N + 85056); etc.

G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381 [math.GM], 2017.

The first column of this triangle is A290030; alternating entries of the first column give A260326. See also triangle A290053, whose columns are A000012-A000096, A290061-A290071, A290127-A290723, etc.

A290061 a(n) = (1/24)(n + 3)(3n^3 + 5n^2 - 6*n + 16).

Original entry on oeis.org

3, 10, 31, 77, 162, 303, 520, 836, 1277, 1872, 2653, 3655, 4916, 6477, 8382, 10678, 13415, 16646, 20427, 24817, 29878, 35675, 42276, 49752, 58177, 67628, 78185, 89931, 102952, 117337, 133178, 150570, 169611, 190402, 213047, 237653, 264330, 293191, 324352, 357932

Offset: 1

Gregory Gerard Wojnar, Jul 19 2017

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

Column 3 of A290053.

Table[(1/24)(n+3)(3n^3+5n^2-6n+16),{n,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{3,10,31,77,162},40] (* Harvey P. Dale, Oct 29 2018 *)

Vec(x*(3 - 5*x + 11*x^2 - 8*x^3 + 2*x^4) / (1 - x)^5 + O(x^50)) \\ Colin Barker, Jul 20 2017

vector(50,n,(n+3)*(3*n^3+5*n^2-6*n+16)/24) \\ Derek Orr, Jul 24 2017

From Colin Barker, Jul 20 2017: (Start)
G.f.: x*(3 - 5*x + 11*x^2 - 8*x^3 + 2*x^4) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
(End)

A290071 a(n) = (1/48)n(n+5)^2(1n^3 + 7n^2 + 16n + 28).

Original entry on oeis.org

0, 39, 196, 664, 1809, 4250, 8954, 17346, 31434, 53949, 88500, 139744, 213571, 317304, 459914, 652250, 907284, 1240371, 1669524, 2215704, 2903125, 3759574, 4816746, 6110594, 7681694, 9575625, 11843364, 14541696, 17733639, 21488884, 25884250, 31004154, 36941096

Offset: 0

Gregory Gerard Wojnar, Jul 19 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

This is the negation of column 4 in triangle A290053.

LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,39,196,664,1809,4250,8954},40] (* Harvey P. Dale, Nov 15 2022 *)

concat(0, Vec(x*(39 - 77*x + 111*x^2 - 88*x^3 + 36*x^4 - 6*x^5) / (1 - x)^7 + O(x^50))) \\ Colin Barker, Jul 20 2017

vector(50,n,n*(n+5)^2*(n^3+7*n^2+16*n+28)/48) \\ Derek Orr, Jul 24 2017

From Colin Barker, Jul 20 2017: (Start)
G.f.: x*(39 - 77*x + 111*x^2 - 88*x^3 + 36*x^4 - 6*x^5) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 6.
(End)

A290127 a(n) = (1/5760)(n + 5)(15n^7 + 225n^6 + 1265n^5 + 3707n^4 + 7120n^3 + 4900n^2 - 6480*n + 27648).

Original entry on oeis.org

40, 252, 1457, 6168, 20773, 59279, 149271, 340821, 719187, 1422247, 2663718, 4763315, 8185110, 13585456, 21871946, 34274982, 52433634, 78497574, 115246975, 166232370, 235936571, 329960853, 455237713, 620272619, 835417269, 1113176985, 1468554972, 1919436277

Offset: 1

Gregory Gerard Wojnar, Jul 20 2017

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

This is column 5 of triangle A290053.

LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{40,252,1457,6168,20773,59279,149271,340821,719187},30] (* Harvey P. Dale, Jul 17 2024 *)

Vec(x*(40 - 108*x + 629*x^2 - 1233*x^3 + 1585*x^4 - 1306*x^5 + 666*x^6 - 192*x^7 + 24*x^8) / (1 - x)^9 + O(x^30)) \\ Colin Barker, Aug 09 2017

G.f.: x*(40 - 108*x + 629*x^2 - 1233*x^3 + 1585*x^4 - 1306*x^5 + 666*x^6 - 192*x^7 + 24*x^8) / (1 - x)^9. - Colin Barker, Aug 09 2017

A290723 a(n) = (1/11520) * n(n+7)^2 (3n^7 + 83n^6 + 961n^5 + 6201n^4 + 24708n^3 + 60700n^2 + 87968*n + 85056).

Original entry on oeis.org

0, 1476, 11772, 61595, 249986, 846306, 2495961, 6601035, 15978570, 35938992, 75976077, 152318826, 291665618, 536502980, 952506198, 1638627738, 2740602996, 4468742196, 7121033250, 11112754029, 17013984714, 25596622646, 37892734319, 55266332805, 79500944910

Offset: 0

Gregory Gerard Wojnar, Aug 09 2017

Colin Barker, 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).

This is the negation of column 6 of triangle A290053.

CoefficientList[Series[x (1476 - 4464 x + 13283 x^2 - 23639 x^3 + 28885 x^4 - 24502 x^5 + 14202 x^6 - 5376 x^7 + 1200 x^8 - 120 x^9)/(1 - x)^11, {x, 0, 24}], x] (* Michael De Vlieger, Aug 09 2017 *)

concat(0, Vec(x*(1476 - 4464*x + 13283*x^2 - 23639*x^3 + 28885*x^4 - 24502*x^5 + 14202*x^6 - 5376*x^7 + 1200*x^8 - 120*x^9) / (1 - x)^11 + O(x^30))) \\ Colin Barker, Aug 09 2017

G.f.: x*(1476 - 4464*x + 13283*x^2 - 23639*x^3 + 28885*x^4 - 24502*x^5 + 14202*x^6 - 5376*x^7 + 1200*x^8 - 120*x^9) / (1 - x)^11. - Colin Barker, Aug 09 2017

cp's OEIS Frontend

A290761 Irregular triangle, read by rows, of coefficients of polynomials that are the "nonstandard" factor of polynomials yielding the columns (up to sign) of triangle A290053, beginning with column 3.

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A290061 a(n) = (1/24)*(n + 3)*(3*n^3 + 5*n^2 - 6*n + 16).

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A290071 a(n) = (1/48)*n*(n+5)^2*(1*n^3 + 7*n^2 + 16*n + 28).

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PARI

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A290127 a(n) = (1/5760)*(n + 5)*(15*n^7 + 225*n^6 + 1265*n^5 + 3707*n^4 + 7120*n^3 + 4900*n^2 - 6480*n + 27648).

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A290723 a(n) = (1/11520) * n*(n+7)^2 * (3*n^7 + 83*n^6 + 961*n^5 + 6201*n^4 + 24708*n^3 + 60700*n^2 + 87968*n + 85056).

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Mathematica

PARI

Formula

A290061 a(n) = (1/24)(n + 3)(3n^3 + 5n^2 - 6*n + 16).

A290071 a(n) = (1/48)n(n+5)^2(1n^3 + 7n^2 + 16n + 28).

A290127 a(n) = (1/5760)(n + 5)(15n^7 + 225n^6 + 1265n^5 + 3707n^4 + 7120n^3 + 4900n^2 - 6480*n + 27648).

A290723 a(n) = (1/11520) * n(n+7)^2 (3n^7 + 83n^6 + 961n^5 + 6201n^4 + 24708n^3 + 60700n^2 + 87968*n + 85056).