A318557 -id:A318557 - cp's OEIS frontend

A318597 Number of n-member subsets of [9*n] whose elements sum to a multiple of nine.

1, 1, 17, 327, 6545, 135751, 2869719, 61474519, 1329890705, 28987537806, 635627275767, 14005614014756, 309847743790935, 6878045467021470, 153121753939078375, 3417374882384868702, 76435265515004093841, 1712879663609111933495, 38450113512250685307822

Offset: 0

Alois P. Heinz, Aug 29 2018

nonn

			a(2) = 17: {1,8}, {1,17}, {2,7}, {2,16}, {3,6}, {3,15}, {4,5}, {4,14}, {5,13}, {6,12}, {7,11}, {8,10}, {9,18}, {10,17}, {11,16}, {12,15}, {13,14}.

Alois P. Heinz, Table of n, a(n) for n = 0..735

Column k=9 of A318557.

A318598 Number of n-member subsets of [10*n] whose elements sum to a multiple of ten.

Original entry on oeis.org

1, 1, 18, 406, 9158, 211880, 5005980, 119877472, 2898762854, 70625252863, 1731030733840, 42634215112710, 1054285960975268, 26159486052576800, 650961394904288192, 16239221627803363280, 405994987399334500710, 10169699086799914175514, 255172150241034559336782

Offset: 0

Alois P. Heinz, Aug 29 2018

nonn

			a(2) = 18: {1,9}, {1,19}, {2,8}, {2,18}, {3,7}, {3,17}, {4,6}, {4,16}, {5,15}, {6,14}, {7,13}, {8,12}, {9,11}, {10,20}, {11,19}, {12,18}, {13,17}, {14,16}.

Alois P. Heinz, Table of n, a(n) for n = 0..710

Column k=10 of A318557.

A318624 Number of 3-member subsets of [3*n] whose elements sum to a multiple of n.

Original entry on oeis.org

0, 1, 10, 30, 55, 91, 138, 190, 253, 327, 406, 496, 597, 703, 820, 948, 1081, 1225, 1380, 1540, 1711, 1893, 2080, 2278, 2487, 2701, 2926, 3162, 3403, 3655, 3918, 4186, 4465, 4755, 5050, 5356, 5673, 5995, 6328, 6672, 7021, 7381, 7752, 8128, 8515, 8913, 9316

Offset: 0

Alois P. Heinz, Aug 30 2018

			a(1) = 1: {1,2,3}.
a(2) = 10: {1,2,3}, {1,2,5}, {1,3,4}, {1,3,6}, {1,4,5}, {1,5,6}, {2,3,5}, {2,4,6}, {3,4,5}, {3,5,6}.
a(3) = 30: {1,2,3}, {1,2,6}, {1,2,9}, {1,3,5}, {1,3,8}, {1,4,7}, {1,5,6}, {1,5,9}, {1,6,8}, {1,8,9}, {2,3,4}, {2,3,7}, {2,4,6}, {2,4,9}, {2,5,8}, {2,6,7}, {2,7,9}, {3,4,5}, {3,4,8}, {3,5,7}, {3,6,9}, {3,7,8}, {4,5,6}, {4,5,9}, {4,6,8}, {4,8,9}, {5,6,7}, {5,7,9}, {6,7,8}, {7,8,9}.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Row n=3 of A318557.

LinearRecurrence[{2,-1,1,-2,1},{0,1,10,30,55,91},50] (* Harvey P. Dale, Mar 27 2019 *)

G.f.: -x*(3*x^4+4*x^3+11*x^2+8*x+1)/((x^2+x+1)*(x-1)^3).
a(n) = 2*a(n-1) -a(n-2) +a(n-3) -2*a(n-4) +a(n-5) for n>5.
3*a(n) = 5+2*A099837(n)+27*n*(n-1)/2 for n>0. - R. J. Mathar, Sep 02 2018

A318625 Number of 4-member subsets of [4*n] whose elements sum to a multiple of n.

Original entry on oeis.org

0, 1, 38, 165, 460, 969, 1782, 2925, 4508, 6545, 9158, 12341, 16236, 20825, 26262, 32509, 39740, 47905, 57190, 67525, 79116, 91881, 106038, 121485, 138460, 156849, 176902, 198485, 221868, 246905, 273878, 302621, 333436, 366145, 401062, 437989, 477260, 518665

Offset: 0

Alois P. Heinz, Aug 30 2018

			a(2) = 38: {1,2,3,4}, {1,2,3,6}, {1,2,3,8}, {1,2,4,5}, {1,2,4,7}, {1,2,5,6}, {1,2,5,8}, {1,2,6,7}, {1,2,7,8}, {1,3,4,6}, {1,3,4,8}, {1,3,5,7}, {1,3,6,8}, {1,4,5,6}, {1,4,5,8}, {1,4,6,7}, {1,4,7,8}, {1,5,6,8}, {1,6,7,8}, {2,3,4,5}, {2,3,4,7}, {2,3,5,6}, {2,3,5,8}, {2,3,6,7}, {2,3,7,8}, {2,4,5,7}, {2,4,6,8}, {2,5,6,7}, {2,5,7,8}, {3,4,5,6}, {3,4,5,8}, {3,4,6,7}, {3,4,7,8}, {3,5,6,8}, {3,6,7,8}, {4,5,6,7}, {4,5,7,8}, {5,6,7,8}.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1).

Row n=4 of A318557.

G.f.: x*(4*x^7+27*x^6+100*x^5+123*x^4+132*x^3+89*x^2+36*x+1) / ((x^2+1) *(x+1)^2 *(x-1)^4).

A318626 Number of 5-member subsets of [5*n] whose elements sum to a multiple of n.

Original entry on oeis.org

0, 1, 126, 1001, 3876, 10630, 23751, 46376, 82251, 135751, 211880, 316251, 455126, 635376, 864501, 1150630, 1502501, 1929501, 2441626, 3049501, 3764380, 4598126, 5563251, 6672876, 7940751, 9381255, 11009376, 12840751, 14891626, 17178876, 19720005, 22533126

Offset: 0

Alois P. Heinz, Aug 30 2018

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,1,-4,6,-4,1).

Row n=5 of A318557.

LinearRecurrence[{4,-6,4,-1,1,-4,6,-4,1},{0,1,126,1001,3876,10630,23751,46376,82251,135751},40] (* Harvey P. Dale, Jul 13 2019 *)

G.f.: -x*(5*x^8+106*x^7+527*x^6+608*x^5+629*x^4+624*x^3+503*x^2+122*x+1) / ((x^4+x^3+x^2+x+1) *(x-1)^5).

a(n) = (-250*n +216/5 + 625*n^4 -1250*n^3 +875*n^2)/24 +4* b(n)/5 for n>=1, where b(n) = b(n-5) = 4, -1, -1, -1, -1, 4, -1, -1, -1,.. 5-periodic for n>=0. - R. J. Mathar, Apr 17 2024

A318627 Number of 6-member subsets of [6*n] whose elements sum to a multiple of n.

Original entry on oeis.org

0, 1, 452, 6198, 33594, 118755, 324516, 749398, 1533686, 2869719, 5005980, 8259888, 13019358, 19757815, 29033576, 41507700, 57939438, 79208745, 106307256, 140364532, 182635562, 234531357, 297600576, 373566942, 464304450, 571880029, 698523980, 846678498

Offset: 0

Alois P. Heinz, Aug 30 2018

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,4,-3,-3,4,1,-1,-3,1,2,-1).

Row n=6 of A318557.

G.f.: x*(6*x^14 +450*x^13 +5286*x^12 +20787*x^11 +46696*x^10 +72381*x^9 +88225*x^8 +92230*x^7 +88135*x^6 +72457*x^5 +46726*x^4 +20749*x^3 +5293*x^2 +450*x+1) / ((x^2-x+1) *(x^2+x+1)^2 *(x+1)^3 *(x-1)^6).

A318628 Number of 7-member subsets of [7*n] whose elements sum to a multiple of n.

Original entry on oeis.org

0, 1, 1716, 38760, 296010, 1344904, 4496388, 12271518, 28989675, 61474519, 119877472, 218618940, 377447148, 622614630, 988172374, 1517381580, 2264243157, 3295144749, 4690625500, 6547258432, 8979650478, 12122560170, 16133132940, 21193254160, 27512019711

Offset: 0

Alois P. Heinz, Aug 30 2018

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1,1,-6,15,-20,15,-6,1).

Row n=7 of A318557.

G.f.: -x*(7*x^12 +1674*x^11 +28569*x^10 +89050*x^9 +116029*x^8 +117612*x^7 +117655*x^6 +117648*x^5 +115939*x^4 +89170*x^3 +28479*x^2 +1710*x+1) / ((x^6+x^5+x^4+x^3+x^2+x+1) *(x-1)^7).

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) + a(n-7) - 6*a(n-8) + 15*a(n-9) - 20*a(n-10) + 15*a(n-11) - 6*a(n-12) + a(n-13). - Wesley Ivan Hurt, Mar 25 2023

A318629 Number of 8-member subsets of [8*n] whose elements sum to a multiple of n.

Original entry on oeis.org

0, 1, 6470, 245157, 2630044, 15380937, 62893270, 202927725, 553275192, 1329890705, 2898762854, 5843355957, 11050100956, 19813501785, 33963673590, 56017460733, 89356455544, 138432467745, 209004466054, 308406729925, 445853163548, 632776453353, 883208213270

Offset: 0

Alois P. Heinz, Aug 30 2018

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-4,-4,11,-8,0,8,-10,0,8,0,-10,8,0,-8,11,-4,-4,4,-1).

Row n=8 of A318557.

G.f.: x*(8*x^19 +6403*x^18 +219614*x^17 +1674059*x^16 +5870564*x^15 +12792670*x^14 +20763588*x^13 +27678950*x^12 +31887236*x^11 +33320524*x^10 +33336672*x^9 +31872212*x^8 +27691884*x^7 +20753602*x^6 +12799164*x^5 +5867258*x^4 +1675300*x^3 +219281*x^2 +6466*x+1) / ((x^4+1) *(x^2+1)^2 *(x+1)^4 *(x-1)^8).

A318630 Number of 9-member subsets of [9*n] whose elements sum to a multiple of n.

Original entry on oeis.org

0, 1, 24310, 1562331, 23535820, 177232627, 886322982, 3381098545, 10639125640, 28987537806, 70625252863, 157366449604, 325949658015, 635299897870, 1176174344125, 2083664997876, 3553074951283, 5859727868575, 9383313282102, 14637413449945, 22304900540890

Offset: 0

Alois P. Heinz, Aug 30 2018

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,22,-27,36,-42,36,-27,23,-21,21,-23,27,-36,42,-36,27,-22,15,-6,1).

Row n=9 of A318557.

G.f.: -x*(9*x^20 +24256*x^19 +1416550*x^18 +14526732*x^17 +58916595*x^16 +142251894*x^15 +245168253*x^14 +328503852*x^13 +372895647*x^12 +386000952*x^11 +387373441*x^10 +386005450*x^9 +372890715*x^8 +328505514*x^7 +245169837*x^6 +142249572*x^5 +58917879*x^4 +14526462*x^3 +1416486*x^2 +24304*x+1) / ((x^6+x^3+1) *(x^2+x+1)^3 *(x-1)^9).

A318631 Number of 10-member subsets of [10*n] whose elements sum to a multiple of n.

Original entry on oeis.org

0, 1, 92252, 10015005, 211911256, 2054455670, 12565647510, 56672074888, 205811431514, 635627275767, 1731030733840, 4263421511271, 9672347764772, 20492404684400, 40975640875026, 77970286548270, 142128053955030, 249524848105533, 423793107835284, 698878653009537

Offset: 0

Alois P. Heinz, Aug 30 2018

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-2,-12,17,9,-32,10,29,-29,-9,28,-7,-5,-5,-7,28,-9,-29,29,10,-32,9,17,-12,-2,4,-1).

Row n=10 of A318557.

G.f.: x*(10*x^26 +92338*x^25 +9646514*x^24 +172035984*x^23 +1227946110*x^22 +4890258085*x^21 +12890246964*x^20 +25218221137*x^19 +40000080160*x^18 +54781619237*x^17 +67109913895*x^16 +75109489209*x^15 +78762565667*x^14 +79655768180*x^13 +78762567575*x^12 +75109489423*x^11 +67109913979*x^10 +54781618619*x^9 +40000079524*x^8 +25218220679*x^7 +12890245939*x^6 +4890259109*x^5 +1227947663*x^4 +172035752*x^3 +9645999*x^2 +92248*x+1) / ((x^4-x^3+x^2-x+1) *(x^4+x^3+x^2+x+1)^2 *(x+1)^5 *(x-1)^10).

cp's OEIS Frontend

A318597 Number of n-member subsets of [9*n] whose elements sum to a multiple of nine.

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A318598 Number of n-member subsets of [10*n] whose elements sum to a multiple of ten.

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A318624 Number of 3-member subsets of [3*n] whose elements sum to a multiple of n.

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A318625 Number of 4-member subsets of [4*n] whose elements sum to a multiple of n.

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A318626 Number of 5-member subsets of [5*n] whose elements sum to a multiple of n.

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A318627 Number of 6-member subsets of [6*n] whose elements sum to a multiple of n.

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A318628 Number of 7-member subsets of [7*n] whose elements sum to a multiple of n.

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A318629 Number of 8-member subsets of [8*n] whose elements sum to a multiple of n.

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A318630 Number of 9-member subsets of [9*n] whose elements sum to a multiple of n.

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A318631 Number of 10-member subsets of [10*n] whose elements sum to a multiple of n.

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