A351145
Triangle T(n,m) read by rows: Sum_{k=1..m} binomial(2*n, n+k)*d(k), m<=n, with d(k)=A000005(k).
Original entry on oeis.org
1, 4, 6, 15, 27, 29, 56, 112, 128, 131, 210, 450, 540, 570, 572, 792, 1782, 2222, 2420, 2444, 2448, 3003, 7007, 9009, 10101, 10283, 10339, 10341, 11440, 27456, 36192, 41652, 42772, 43252, 43284, 43288, 43758, 107406, 144534, 170238, 176358, 179622, 179928, 180000, 180003
Offset: 1
The triangle begins:
1;
4, 6;
15, 27, 29;
56, 112, 128, 131;
210, 450, 540, 570, 572;
792, 1782, 2222, 2420, 2444, 2448;
3003, 7007, 9009, 10101, 10283, 10339, 10341;
11440, 27456, 36192, 41652, 42772, 43252, 43284, 43288;
- D. E. Knuth, The Art of Computer Programming Second Edition. Vol. 3, Sorting and Searching. Chapter 5.2.2 Sorting by Exchanging, pages 138, 637 (answer to exercise 52). Addison-Wesley, Reading, MA, 1998.
-
T[n_, m_] := Sum[Binomial[2*n, n + k] * DivisorSigma[0, k], {k, 1, m}]; Table[T[n, m], {n, 1, 9}, {m, 1, n}] // Flatten (* Amiram Eldar, Feb 02 2022 *)
-
for(n=1,10,for(m=1,n,my(s=sum(t=1,m,binomial(2*n,n+t)*numdiv(t)));print1(s,", ")))
A356338
a(n) = Sum_{k=1..n} binomial(2*n, n-k) * sigma(k).
Original entry on oeis.org
1, 7, 37, 179, 826, 3703, 16283, 70619, 303121, 1290682, 5460511, 22981019, 96296552, 402024497, 1673116072, 6944105579, 28752345362, 118801061059, 489959398840, 2017339105514, 8293732341134, 34051489445365, 139634028015269, 571955737066307, 2340402722605976, 9567794393004816
Offset: 1
-
Table[Sum[Binomial[2*n, n-k]*DivisorSigma[1, k], {k, 1, n}], {n, 1, 30}]
-
a(n) = sum(k=1, n, binomial(2*n, n-k) * sigma(k)); \\ Michel Marcus, Aug 05 2022
A356339
a(n) = Sum_{k=1..n} binomial(2*n, n-k) * sigma_2(k).
Original entry on oeis.org
1, 9, 55, 297, 1496, 7215, 33783, 154825, 698077, 3107424, 13690161, 59802471, 259377080, 1118176887, 4795381640, 20472223529, 87051685546, 368857919085, 1558036408998, 6562564601592, 27571934249754, 115574440020477, 483444570596465, 2018365519396135, 8411811012694246
Offset: 1
-
Table[Sum[Binomial[2*n, n-k]*DivisorSigma[2, k], {k, 1, n}], {n, 1, 30}]
-
a(n) = sum(k=1, n, binomial(2*n, n-k) * sigma(k, 2)); \\ Michel Marcus, Aug 05 2022
A356340
a(n) = Sum_{k=1..n} binomial(2*n, n-k) * phi(k), where phi is the Euler totient function.
Original entry on oeis.org
1, 5, 23, 102, 444, 1909, 8133, 34404, 144714, 605920, 2527348, 10507978, 43569096, 180219699, 743907057, 3065019864, 12607648238, 51783970314, 212412697368, 870249992168, 3561502879100, 14560944187796, 59476980459794, 242741090637012, 989921853052930, 4034101567907172
Offset: 1
-
Table[Sum[Binomial[2*n, n-k]*EulerPhi[k], {k, 1, n}], {n, 1, 30}]
-
a(n) = sum(k=1, n, binomial(2*n, n-k) * eulerphi(k)); \\ Michel Marcus, Aug 05 2022
A356341
a(n) = Sum_{k=1..n} binomial(2*n, k) * sigma(k).
Original entry on oeis.org
2, 22, 131, 806, 3607, 20395, 84254, 422230, 1842359, 8616007, 33843614, 173724659, 676938316, 2983855666, 12806013721, 57981927158, 223432922515, 1040923729567, 4004885305320, 18277809794671, 75668287229078, 317458937099194, 1215454524390767, 5785782106653667
Offset: 1
-
Table[Sum[Binomial[2*n, k]*DivisorSigma[1, k], {k, 1, n}], {n, 1, 30}]
-
a(n) = sum(k=1, n, binomial(2*n, k) * sigma(k)); \\ Michel Marcus, Aug 05 2022
A356342
a(n) = Sum_{k=1..n} binomial(2*n, k) * sigma_2(k).
Original entry on oeis.org
2, 34, 281, 2178, 12397, 79729, 398932, 2224354, 10959221, 56341309, 255685080, 1334248401, 5892916876, 28082515768, 127714609741, 604178948098, 2590365128017, 12284868071365, 52160408294826, 241445420212893, 1049251819301974, 4674022621994716, 19563451165603647
Offset: 1
-
Table[Sum[Binomial[2*n, k]*DivisorSigma[2, k], {k, 1, n}], {n, 1, 30}]
-
a(n) = sum(k=1, n, binomial(2*n, k) * sigma(k, 2)); \\ Michel Marcus, Aug 05 2022
A356344
a(n) = Sum_{k=1..n} binomial(2*k, k) * sigma(k).
Original entry on oeis.org
2, 20, 100, 590, 2102, 13190, 40646, 233696, 865756, 4191364, 12656548, 88372916, 233981316, 1196779716, 4919600196, 23553092286, 65558004246, 419488280946, 1126393556946, 6915947767386, 24140199749466, 99887762443386, 297490099905786, 2232346320891786, 6151075120462098
Offset: 1
-
Table[Sum[Binomial[2*k, k]*DivisorSigma[1, k], {k, 1, n}], {n, 1, 30}]
-
a(n) = sum(k=1, n, binomial(2*k, k) * sigma(k)); \\ Michel Marcus, Aug 05 2022
A356345
a(n) = Sum_{k=1..n} binomial(2*k, k) * sigma_2(k).
Original entry on oeis.org
2, 32, 232, 1702, 8254, 54454, 226054, 1320004, 5744424, 29762704, 115825408, 683698168, 2451800168, 12480950168, 52811505368, 257779918358, 934525722158, 5063712283658, 17858697779258, 93122902514978, 362251839734978, 1645752207604178, 6009470493232178, 33419933623867178
Offset: 1
-
Table[Sum[Binomial[2*k, k]*DivisorSigma[2, k], {k, 1, n}], {n, 1, 30}]
-
a(n) = sum(k=1, n, binomial(2*k, k) * sigma(k, 2)); \\ Michel Marcus, Aug 05 2022
A356372
a(n) = Sum_{k=1..n} binomial(2*n, k) * A000005(k).
Original entry on oeis.org
2, 16, 76, 386, 1474, 7349, 26807, 121964, 487068, 2105087, 7486505, 37278746, 133488216, 550615531, 2263230587, 9856735046, 35168418266, 160420872009, 573578559659, 2582163925152, 10333237435638, 41122278086361, 146621866522577, 712999981650663, 2702556741014621
Offset: 1
-
Table[Sum[Binomial[2*n, k]*DivisorSigma[0, k], {k, 1, n}], {n, 1, 30}]
-
a(n) = sum(k=1, n, binomial(2*n, k) * numdiv(k)); \\ Michel Marcus, Aug 05 2022
A356373
a(n) = Sum_{k=1..n} binomial(2*k, k) * A000005(k).
Original entry on oeis.org
2, 14, 54, 264, 768, 4464, 11328, 62808, 208668, 947692, 2358556, 18583492, 39384692, 199851092, 820321172, 3825723122, 8492935562, 62943747362, 133634274962, 960713447882, 3113744945642, 11530140800522, 27997002255722, 285977831720522, 665209651033778, 2648883782826194
Offset: 1
-
Table[Sum[Binomial[2*k, k]*DivisorSigma[0, k], {k, 1, n}], {n, 1, 30}]
-
a(n) = sum(k=1, n, binomial(2*k, k) * numdiv(k)); \\ Michel Marcus, Aug 05 2022
Showing 1-10 of 10 results.
Comments