A282011 -id:A282011 - cp's OEIS frontend

A282078 Number of 10-element subsets of [n+10] having an even sum.

0, 5, 30, 140, 490, 1491, 3976, 9696, 21816, 46126, 92252, 176232, 323092, 571802, 980232, 1633984, 2655224, 4217499, 6560554, 10014004, 15021006, 22174581, 32253936, 46278336, 65560976, 91786604, 127089144, 174160784, 236361064, 317866884, 423822512

Offset: 0

Alois P. Heinz, Feb 05 2017

			a(1) = 5: {1,2,3,4,5,6,7,8,9,11}, {1,2,3,4,5,6,7,9,10,11}, {1,2,3,4,5,7,8,9,10,11}, {1,2,3,5,6,7,8,9,10,11}, {1,3,4,5,6,7,8,9,10,11}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-10,-10,50,-34,-66,110,0,-110,66,34,-50,10,10,-6,1).

Column k=10 of A282011.

concat(0, Vec(-x*(x^4+10*x^2+5)/((1+x)^5*(x-1)^11) + O(x^30))) \\ Colin Barker, Feb 06 2017

G.f.: -x*(x^4+10*x^2+5)/((1+x)^5*(x-1)^11).

a(n) = (-2735775*(-1+(-1)^n) - 45*(-344851 + 56595*(-1)^n)*n + (22908402-803250*(-1)^n)*n^2 - 50*(-325607+2079*(-1)^n)*n^3 + (6781885-4725*(-1)^n)*n^4 + 1802220*n^5 + 315546*n^6 + 36300*n^7 + 2640*n^8 + 110*n^9 + 2*n^10) / 14515200. - Colin Barker, Feb 06 2017

A282079 Number of n-element subsets of [n+2] having an even sum.

Original entry on oeis.org

1, 1, 2, 6, 9, 9, 12, 20, 25, 25, 30, 42, 49, 49, 56, 72, 81, 81, 90, 110, 121, 121, 132, 156, 169, 169, 182, 210, 225, 225, 240, 272, 289, 289, 306, 342, 361, 361, 380, 420, 441, 441, 462, 506, 529, 529, 552, 600, 625, 625, 650, 702, 729, 729, 756, 812, 841

Offset: 0

Alois P. Heinz, Feb 05 2017

			a(3) = 6: {1,2,3}, {1,2,5}, {1,3,4}, {1,4,5}, {2,3,5}, {3,4,5}.
a(4) = 9: {1,2,3,4}, {1,2,3,6}, {1,2,4,5}, {1,2,5,6}, {1,3,4,6}, {1,4,5,6}, {2,3,4,5}, {2,3,5,6}, {3,4,5,6}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1).

Cf. A282011.

Vec(-(x^4-2*x^3+4*x^2-2*x+1) / ((x^2+1)^2*(x-1)^3) + O(x^90)) \\ Colin Barker, Feb 06 2017

G.f.: -(x^4-2*x^3+4*x^2-2*x+1)/((x^2+1)^2*(x-1)^3).
a(n) = A282011(n+2,n).
a(n) = (2*(1+n)*(2+n) - i*(-i)^n*((1+2*i)+(1+i)*n) + i^n*((2+i)+(1+i)*n))/8 where i=sqrt(-1). - Colin Barker, Feb 06 2017

A282080 Number of n-element subsets of [n+4] having an even sum.

Original entry on oeis.org

1, 2, 6, 19, 38, 60, 100, 170, 255, 350, 490, 693, 924, 1176, 1512, 1956, 2445, 2970, 3630, 4455, 5346, 6292, 7436, 8814, 10283, 11830, 13650, 15785, 18040, 20400, 23120, 26248, 29529, 32946, 36822, 41211, 45790, 50540, 55860, 61810, 67991, 74382, 81466, 89309

Offset: 0

Alois P. Heinz, Feb 05 2017

			a(2) = 6: {1,3}, {1,5}, {2,4}, {2,6}, {3,5}, {4,6}.
a(3) = 19: {1,2,3}, {1,2,5}, {1,2,7}, {1,3,4}, {1,3,6}, {1,4,5}, {1,4,7}, {1,5,6}, {1,6,7}, {2,3,5}, {2,3,7}, {2,4,6}, {2,5,7}, {3,4,5}, {3,4,7}, {3,5,6}, {3,6,7}, {4,5,7}, {5,6,7}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-13,25,-38,46,-46,38,-25,13,-5,1).

Cf. A282011.

CoefficientList[Series[-(x^2 - x + 1)*(x^4 - 2*x^3 + 6*x^2 - 2*x + 1)/((x^2 + 1)^3*(x - 1)^5), {x, 0, 50}], x] (* Wesley Ivan Hurt, Jan 01 2024 *)

Vec(-(x^2-x+1)*(x^4-2*x^3+6*x^2-2*x+1)/((x^2+1)^3*(x-1)^5) + O(x^90)) \\ Colin Barker, Feb 06 2017

G.f.: -(x^2-x+1)*(x^4-2*x^3+6*x^2-2*x+1)/((x^2+1)^3*(x-1)^5).
a(n) = A282011(n+4,n).
a(n) = (2*(1+n)*(2+n)*(3+n)*(4+n) + 3*(-i*(-i)^n*((3+8*i) + (4+6*i)*n + (1+i)*n^2) + i^n*((8+3*i) + (6+4*i)*n + (1+i)*n^2)))/96 where i=sqrt(-1). - Colin Barker, Feb 06 2017

A282081 Number of n-element subsets of [n+5] having an even sum.

Original entry on oeis.org

1, 3, 9, 28, 66, 126, 226, 396, 651, 1001, 1491, 2184, 3108, 4284, 5796, 7752, 10197, 13167, 16797, 21252, 26598, 32890, 40326, 49140, 59423, 71253, 84903, 100688, 118728, 139128, 162248, 188496, 218025, 250971, 287793, 329004, 374794, 425334, 481194, 543004

Offset: 0

Alois P. Heinz, Feb 05 2017

			a(0) = 1: {}.
a(1) = 3: {2}, {4}, {6}.
a(2) = 9: {1,3}, {1,5}, {1,7}, {2,4}, {2,6}, {3,5}, {3,7}, {4,6}, {5,7}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-18,38,-63,84,-92,84,-63,38,-18,6,-1).

Cf. A282011.

LinearRecurrence[{6,-18,38,-63,84,-92,84,-63,38,-18,6,-1},{1,3,9,28,66,126,226,396,651,1001,1491,2184},40] (* Harvey P. Dale, Sep 30 2019 *)

Vec((x^2-x+1)*(x^4-2*x^3+6*x^2-2*x+1) / ((x^2+1)^3*(x-1)^6) + O(x^60)) \\ Colin Barker, Feb 06 2017

G.f.: (x^2-x+1)*(x^4-2*x^3+6*x^2-2*x+1)/((x^2+1)^3*(x-1)^6).
a(n) = A282011(n+5,n).
a(n) = (1+n)*(2+n)*(3+n)*(4+n)*(5+n)/240 + ((-i)^n+i^n)*(8+6*n+n^2)/32 where i=sqrt(-1). - Colin Barker, Feb 06 2017

A282082 Number of n-element subsets of [n+6] having an even sum.

Original entry on oeis.org

1, 3, 12, 44, 110, 226, 452, 868, 1519, 2485, 3976, 6216, 9324, 13524, 19320, 27192, 37389, 50391, 67188, 88660, 115258, 147862, 188188, 237692, 297115, 367913, 452816, 554064, 672792, 811240, 973488, 1162800, 1380825, 1630827, 1918620, 2248764, 2623558

Offset: 0

Alois P. Heinz, Feb 05 2017

			a(0) = 1: {}.
a(1) = 3: {2}, {4}, {6}.
a(2) = 12: {1,3}, {1,5}, {1,7}, {2,4}, {2,6}, {2,8}, {3,5}, {3,7}, {4,6}, {4,8}, {5,7}, {6,8}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7, -25, 63, -125, 203, -277, 323, -323, 277, -203, 125, -63, 25, -7, 1).

Cf. A282011.

G.f.: -(x^8-4*x^7+16*x^6-28*x^5+38*x^4-28*x^3+16*x^2-4*x+1) / ((x^2+1)^4*(x-1)^7).

a(n) = A282011(n+6,n).

A282083 Number of n-element subsets of [n+7] having an even sum.

Original entry on oeis.org

1, 4, 16, 60, 170, 396, 848, 1716, 3235, 5720, 9696, 15912, 25236, 38760, 58080, 85272, 122661, 173052, 240240, 328900, 444158, 592020, 780208, 1017900, 1315015, 1682928, 2135744, 2689808, 3362600, 4173840, 5147328, 6310128, 7690953, 9321780, 11240400

Offset: 0

Alois P. Heinz, Feb 05 2017

			a(0) = 1: {}.
a(1) = 4: {2}, {4}, {6}, {8}.
a(2) = 16: {1,3}, {1,5}, {1,7}, {1,9}, {2,4}, {2,6}, {2,8}, {3,5}, {3,7}, {3,9}, {4,6}, {4,8}, {5,7}, {5,9}, {6,8}, {7,9}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8, -32, 88, -188, 328, -480, 600, -646, 600, -480, 328, -188, 88, -32, 8, -1).

Cf. A282011.

G.f.: (x^8-4*x^7+16*x^6-28*x^5+38*x^4-28*x^3+16*x^2-4*x+1) / ((x^2+1)^4*(x-1)^8).

a(n) = A282011(n+7,n).

A282084 Number of n-element subsets of [n+8] having an even sum.

Original entry on oeis.org

1, 4, 20, 85, 255, 636, 1484, 3235, 6470, 12120, 21816, 37854, 63090, 101640, 159720, 245322, 367983, 540540, 780780, 1110395, 1554553, 2145572, 2925780, 3945045, 5260060, 6941168, 9076912, 11769100, 15131700, 19302480, 24449808, 30763812, 38454765, 47771700

Offset: 0

Alois P. Heinz, Feb 05 2017

			a(0) = 1: {}.
a(1) = 4: {2}, {4}, {6}, {8}.
a(2) = 20: {1,3}, {1,5}, {1,7}, {1,9}, {2,4}, {2,6}, {2,8}, {2,10}, {3,5}, {3,7}, {3,9}, {4,6}, {4,8}, {4,10}, {5,7}, {5,9}, {6,8}, {6,10}, {7,9}, {8,10}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9, -41, 129, -316, 636, -1084, 1596, -2054, 2326, -2326, 2054, -1596, 1084, -636, 316, -129, 41, -9, 1).

Cf. A282011.

G.f.: -(x^2-x+1)*(x^8-4*x^7+20*x^6-36*x^5+54*x^4-36*x^3+20*x^2-4*x+1) / ((x^2+1)^5*(x-1)^9).

a(n) = A282011(n+8,n).

A282085 Number of n-element subsets of [n+9] having an even sum.

Original entry on oeis.org

1, 5, 25, 110, 365, 1001, 2485, 5720, 12190, 24310, 46126, 83980, 147070, 248710, 408430, 653752, 1021735, 1562275, 2343055, 3453450, 5008003, 7153575, 10079355, 14024400, 19284460, 26225628, 35302540, 47071640, 62203340, 81505820, 105955628, 136719440

Offset: 0

Alois P. Heinz, Feb 05 2017

			a(0) = 1: {}.
a(1) = 5: {2}, {4}, {6}, {8}, {10}.
a(2) = 25: {1,3}, {1,5}, {1,7}, {1,9}, {1,11}, {2,4}, {2,6}, {2,8}, {2,10}, {3,5}, {3,7}, {3,9}, {3,11}, {4,6}, {4,8}, {4,10}, {5,7}, {5,9}, {5,11}, {6,8}, {6,10}, {7,9}, {7,11}, {8,10}, {9,11}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10, -50, 170, -445, 952, -1720, 2680, -3650, 4380, -4652, 4380, -3650, 2680, -1720, 952, -445, 170, -50, 10, -1).

Cf. A282011.

T[n_, k_] := Sum[Binomial[Ceiling[n/2], 2 j] Binomial[Floor[n/2], k - 2 j], {j, 0, Floor[(n + 1)/4]}]; Table[T[n + 9, n], {n, 0, 31}] (* or *) CoefficientList[Series[(x^2 - x + 1) (x^8 - 4 x^7 + 20 x^6 - 36 x^5 + 54 x^4 - 36 x^3 + 20 x^2 - 4 x + 1)/((x^2 + 1)^5 (x - 1)^10), {x, 0, 31}], x] (* Indranil Ghosh, Feb 26 2017 *)

G.f.: (x^2-x+1)*(x^8-4*x^7+20*x^6-36*x^5+54*x^4-36*x^3+20*x^2-4*x+1) / ((x^2+1)^5*(x-1)^10).

a(n) = A282011(n+9,n).

A282086 Number of n-element subsets of [n+10] having an even sum.

Original entry on oeis.org

1, 5, 30, 146, 511, 1491, 3976, 9752, 21942, 46126, 92252, 176484, 323554, 571802, 980232, 1634776, 2656511, 4217499, 6560554, 10016006, 15024009, 22174581, 32253936, 46282704, 65567164, 91786604, 127089144, 174169352, 236372692, 317866884, 423822512

Offset: 0

Alois P. Heinz, Feb 05 2017

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11, -61, 231, -675, 1617, -3287, 5797, -9002, 12430, -15362, 17062, -17062, 15362, -12430, 9002, -5797, 3287, -1617, 675, -231, 61, -11, 1).

Cf. A282011.

G.f.: -(x^4-2*x^3+4*x^2-2*x+1) * (x^8-4*x^7+24*x^6-44*x^5+62*x^4-44*x^3 + 24*x^2-4*x+1) / ((x^2+1)^6*(x-1)^11).

a(n) = A282011(n+10,n).

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A282078 Number of 10-element subsets of [n+10] having an even sum.

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A282079 Number of n-element subsets of [n+2] having an even sum.

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A282080 Number of n-element subsets of [n+4] having an even sum.

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A282081 Number of n-element subsets of [n+5] having an even sum.

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A282082 Number of n-element subsets of [n+6] having an even sum.

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A282083 Number of n-element subsets of [n+7] having an even sum.

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A282084 Number of n-element subsets of [n+8] having an even sum.

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A282085 Number of n-element subsets of [n+9] having an even sum.

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A282086 Number of n-element subsets of [n+10] having an even sum.

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