Colin Barker - cp's OEIS frontend

A317984 Expansion of 140x(1 + 4*x + x^2) / (1 - x)^5.

140, 1260, 5040, 14000, 31500, 61740, 109760, 181440, 283500, 423500, 609840, 851760, 1159340, 1543500, 2016000, 2589440, 3277260, 4093740, 5054000, 6174000, 7470540, 8961260, 10664640, 12600000, 14787500, 17248140, 20003760, 23077040, 26491500, 30271500

Offset: 1

Colin Barker, Aug 13 2018

Seems to be the fourth column of A316387.

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).

Cf. A316387, A317981, A317982, A317983.

Vec(140*x*(1 + 4*x + x^2) / (1 - x)^5 + O(x^40))

PARI
```
a(n) = 35*n^4 + 70*n^3 + 35*n^2
```

G.f.: 140*x*(1 + 4*x + x^2) / (1 - x)^5.
a(n) = 35*n^4 + 70*n^3 + 35*n^2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.

A317983 Expansion of 420x(1 + x)(1 + 10x + x^2) / (1 - x)^6.

Original entry on oeis.org

420, 7140, 41160, 148680, 411180, 955500, 1963920, 3684240, 6439860, 10639860, 16789080, 25498200, 37493820, 53628540, 74891040, 102416160, 137494980, 181584900, 236319720, 303519720, 385201740, 483589260, 601122480, 740468400, 904530900, 1096460820

Offset: 1

Colin Barker, Aug 13 2018

Seems to be the negative of the third column of A316387.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000538, A316387, A317981, A317982, A317984.

Vec(420*x*(1 + x)*(1 + 10*x + x^2) / (1 - x)^6 + O(x^40))

a(n) = 84*n^5 + 210*n^4 + 140*n^3 - 14*n

G.f.: 420*x*(1 + x)*(1 + 10*x + x^2) / (1 - x)^6.
a(n) = 420 * A000538(n).
a(n) = 84*n^5 + 210*n^4 + 140*n^3 - 14*n.
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) for n>6.

A317982 Expansion of 14x(29 + 784x + 1974x^2 + 784x^3 + 29x^4) / (1 - x)^7.

Original entry on oeis.org

406, 13818, 115836, 545860, 1858290, 5124126, 12182968, 25945416, 50745870, 92745730, 160386996, 264896268, 420839146, 646725030, 965662320, 1406064016, 2002403718, 2796022026, 3835983340, 5179983060, 6895305186, 9059830318, 11763094056, 15107395800

Offset: 1

Colin Barker, Aug 13 2018

Seems to be the second column of A316387.

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

Cf. A316387, A317981, A317983, A317984.

LinearRecurrence[{7,-21,35,-35,21,-7,1},{406,13818,115836,545860,1858290,5124126,12182968},30] (* Harvey P. Dale, Nov 15 2022 *)

Vec(14*x*(29 + 784*x + 1974*x^2 + 784*x^3 + 29*x^4) / (1 - x)^7 + O(x^40))

a(n) = 70*n^6 + 210*n^5 + 175*n^4 - 42*n^2 - 7*n

G.f.: 14*x*(29 + 784*x + 1974*x^2 + 784*x^3 + 29*x^4) / (1 - x)^7.
a(n) = 70*n^6 + 210*n^5 + 175*n^4 - 42*n^2 - 7*n.
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>7.

A317981 Expansion of x(125 + 8028x + 42237x^2 + 42272x^3 + 8007x^4 + 132x^5 - x^6) / (1 - x)^8.

Original entry on oeis.org

125, 9028, 110961, 684176, 2871325, 9402660, 25872833, 62572096, 136972701, 276971300, 524988145, 943023888, 1618774781, 2672907076, 4267591425, 6616398080, 9995653693, 14757360516, 21343778801, 30303773200, 42311023965, 58184203748, 78909220801

Offset: 1

Colin Barker, Aug 13 2018

Seems to be the negative of the first column of A316387.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A316387, A317982, A317983, A317984.

LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{125,9028,110961,684176,2871325,9402660,25872833,62572096},30] (* Harvey P. Dale, Dec 29 2024 *)

Vec(x*(125 + 8028*x + 42237*x^2 + 42272*x^3 + 8007*x^4 + 132*x^5 - x^6) / (1 - x)^8 + O(x^40))

a(n) = 20*n^7 + 70*n^6 + 70*n^5 - 28*n^3 - 7*n^2

G.f.: x*(125 + 8028*x + 42237*x^2 + 42272*x^3 + 8007*x^4 + 132*x^5 - x^6) / (1 - x)^8.
a(n) = 20*n^7 + 70*n^6 + 70*n^5 - 28*n^3 - 7*n^2.
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) for n>8.

A316459 Expansion of 30x(1 + x) / (1 - x)^4.

Original entry on oeis.org

30, 150, 420, 900, 1650, 2730, 4200, 6120, 8550, 11550, 15180, 19500, 24570, 30450, 37200, 44880, 53550, 63270, 74100, 86100, 99330, 113850, 129720, 147000, 165750, 186030, 207900, 231420, 256650, 283650, 312480, 343200, 375870, 410550, 447300, 486180

Offset: 1

Colin Barker, Aug 12 2018

Seems to be the third column of A316349.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A316349, A316457, A316458.

Vec(30*x*(1 + x) / (1 - x)^4 + O(x^40))

PARI
```
a(n) = 10*n^3 + 15*n^2 + 5*n
```

G.f.: 30*x*(1 + x) / (1 - x)^4.
a(n) = 30 * A000330(n).
a(n) = 10*n^3 + 15*n^2 + 5*n.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.

A316458 Expansion of 60x(1 + 4*x + x^2) / (1 - x)^5.

Original entry on oeis.org

60, 540, 2160, 6000, 13500, 26460, 47040, 77760, 121500, 181500, 261360, 365040, 496860, 661500, 864000, 1109760, 1404540, 1754460, 2166000, 2646000, 3201660, 3840540, 4570560, 5400000, 6337500, 7392060, 8573040, 9890160, 11353500, 12973500, 14760960

Offset: 1

Colin Barker, Aug 12 2018

Seems to be the negative of the second column of A316349.

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).

Cf. A000537, A316349, A316457, A316459.

Vec(60*x*(1 + 4*x + x^2) / (1 - x)^5 + O(x^40))

PARI
```
a(n) = 15*n^4 + 30*n^3 + 15*n^2
```

G.f.: 60*x*(1 + 4*x + x^2) / (1 - x)^5.
a(n) = 60 * A000537(n).
a(n) = 15*n^4 + 30*n^3 + 15*n^2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.

A316457 Expansion of x(31 + 326x + 336x^2 + 26x^3 + x^4) / (1 - x)^6.

Original entry on oeis.org

31, 512, 2943, 10624, 29375, 68256, 140287, 263168, 459999, 760000, 1199231, 1821312, 2678143, 3830624, 5349375, 7315456, 9821087, 12970368, 16879999, 21680000, 27514431, 34542112, 42937343, 52890624, 64609375, 78318656, 94261887, 112701568, 133919999

Offset: 1

Colin Barker, Aug 12 2018

Seems to be the first column of A316349.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A316349, A316458, A316459.

Vec(x*(31 + 326*x + 336*x^2 + 26*x^3 + x^4) / (1 - x)^6 + O(x^40))

PARI
```
a(n) = 6*n^5 + 15*n^4 + 10*n^3
```

G.f.: x*(31 + 326*x + 336*x^2 + 26*x^3 + x^4) / (1 - x)^6.
a(n) = 6*n^5 + 15*n^4 + 10*n^3.
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) for n>6.

A299335 Expansion of 1 / ((1 - x)^7*(1 + x)^2).

Original entry on oeis.org

1, 5, 17, 45, 103, 211, 399, 707, 1190, 1918, 2982, 4494, 6594, 9450, 13266, 18282, 24783, 33099, 43615, 56771, 73073, 93093, 117481, 146965, 182364, 224588, 274652, 333676, 402900, 483684, 577524, 686052, 811053, 954465, 1118397, 1305129, 1517131, 1757063

Offset: 0

Colin Barker, Feb 07 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-8,0,14,-14,0,8,-5,1).

Cf. A001769, A060099, A299336, A299337, A299338.

CoefficientList[Series[1/((1 - x)^7 (1 + x)^2), {x, 0, 40}], x] (* or *) LinearRecurrence[{5, -8, 0, 14, -14, 0, 8, -5, 1}, {1, 5, 17, 45, 103, 211, 399, 707, 1190}, 41] (* Robert G. Wilson v, Feb 07 2018 *)

Vec(1 / ((1 - x)^7*(1 + x)^2) + O(x^40))

a(n) = (2*n^6 + 54*n^5 + 575*n^4 + 3060*n^3 + 8468*n^2 + 11376*n + 5760) / 5760 for n even.
a(n) = (2*n^6 + 54*n^5 + 575*n^4 + 3060*n^3 + 8468*n^2 + 11286*n + 5355) / 5760 for n odd.
a(n) = 5*a(n-1) - 8*a(n-2) + 14*a(n-4) - 14*a(n-5) + 8*a(n-7) - 5*a(n-8) + a(n-9) for n>8.

A299337 Expansion of 1 / ((1 - x)^7*(1 + x)^5).

Original entry on oeis.org

1, 2, 8, 14, 35, 56, 112, 168, 294, 420, 672, 924, 1386, 1848, 2640, 3432, 4719, 6006, 8008, 10010, 13013, 16016, 20384, 24752, 30940, 37128, 45696, 54264, 65892, 77520, 93024, 108528, 128877, 149226, 175560, 201894, 235543, 269192, 311696, 354200, 407330

Offset: 0

Colin Barker, Feb 07 2018

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Brian Hopkins and Aram Tangboonduangjit, Water Cells in Compositions of 1s and 2s, arXiv:2412.11528 [math.CO], 2024. See p. 3.
Index entries for linear recurrences with constant coefficients, signature (2,4,-10,-5,20,0,-20,5,10,-4,-2,1).

Cf. A001769, A060099, A299335, A299336, A299338.

LinearRecurrence[{2, 4, -10, -5, 20, 0, -20, 5, 10, -4, -2, 1}, {1, 2, 8, 14, 35, 56, 112, 168, 294, 420, 672, 924}, 41] (* Michael De Vlieger, Dec 19 2024 *)

Vec(1 / ((1 - x)^7*(1 + x)^5) + O(x^40))

a(n) = (2*n^6 + 72*n^5 + 1040*n^4 + 7680*n^3 + 30368*n^2 + 60288*n + 46080) / 46080 for n even.
a(n) = (2*n^6 + 72*n^5 + 1010*n^4 + 6960*n^3 + 24278*n^2 + 39048*n + 20790) / 46080 for n odd.
a(n) = 2*a(n-1) + 4*a(n-2) - 10*a(n-3) - 5*a(n-4) + 20*a(n-5) - 20*a(n-7) + 5*a(n-8) + 10*a(n-9) - 4*a(n-10) - 2*a(n-11) + a(n-12) for n>11.

A299338 Expansion of 1 / ((1 - x)^7*(1 + x)^6).

Original entry on oeis.org

1, 1, 7, 7, 28, 28, 84, 84, 210, 210, 462, 462, 924, 924, 1716, 1716, 3003, 3003, 5005, 5005, 8008, 8008, 12376, 12376, 18564, 18564, 27132, 27132, 38760, 38760, 54264, 54264, 74613, 74613, 100947, 100947, 134596, 134596, 177100, 177100, 230230, 230230

Offset: 0

Colin Barker, Feb 07 2018

Same as A000579 but with repeated terms.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1).

Cf. A000579, A001769, A060099, A299335, A299336, A299337.

CoefficientList[Series[1/((1-x)^7(1+x)^6),{x,0,50}],x] (* or *) LinearRecurrence[ {1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1},{1,1,7,7,28,28,84,84,210,210,462,462,924},50] (* Harvey P. Dale, Oct 09 2018 *)

Vec(1 / ((1 - x)^7*(1 + x)^6) + O(x^40))

a(n) = (2*n^6 + 84*n^5 + 1400*n^4 + 11760*n^3 + 51968*n^2 + 112896*n + 92160) / 92160 for n even.
a(n) = (2*n^6 + 72*n^5 + 1010*n^4 + 6960*n^3 + 24278*n^2 + 39048*n + 20790) / 92160 for n odd.
a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - 15*a(n-4) + 15*a(n-5) + 20*a(n-6) - 20*a(n-7) - 15*a(n-8) + 15*a(n-9) + 6*a(n-10) - 6*a(n-11) - a(n-12) + a(n-13) for n>12.

cp's OEIS Frontend

User: Colin Barker

A317984 Expansion of 140*x*(1 + 4*x + x^2) / (1 - x)^5.

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A317983 Expansion of 420*x*(1 + x)*(1 + 10*x + x^2) / (1 - x)^6.

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A317982 Expansion of 14*x*(29 + 784*x + 1974*x^2 + 784*x^3 + 29*x^4) / (1 - x)^7.

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A317981 Expansion of x*(125 + 8028*x + 42237*x^2 + 42272*x^3 + 8007*x^4 + 132*x^5 - x^6) / (1 - x)^8.

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A316459 Expansion of 30*x*(1 + x) / (1 - x)^4.

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A316458 Expansion of 60*x*(1 + 4*x + x^2) / (1 - x)^5.

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A316457 Expansion of x*(31 + 326*x + 336*x^2 + 26*x^3 + x^4) / (1 - x)^6.

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A317984 Expansion of 140x(1 + 4*x + x^2) / (1 - x)^5.

A317983 Expansion of 420x(1 + x)(1 + 10x + x^2) / (1 - x)^6.

A317982 Expansion of 14x(29 + 784x + 1974x^2 + 784x^3 + 29x^4) / (1 - x)^7.

A317981 Expansion of x(125 + 8028x + 42237x^2 + 42272x^3 + 8007x^4 + 132x^5 - x^6) / (1 - x)^8.

A316459 Expansion of 30x(1 + x) / (1 - x)^4.

A316458 Expansion of 60x(1 + 4*x + x^2) / (1 - x)^5.

A316457 Expansion of x(31 + 326x + 336x^2 + 26x^3 + x^4) / (1 - x)^6.