A003236
a(n) = Sum_{k=0..n} (-1)^(n-k) C(n,k)*C((k+1)^2, n).
Original entry on oeis.org
1, 3, 24, 320, 6122, 153762, 4794664, 178788528, 7762727196, 384733667780, 21434922419504, 1326212860090560, 90227121642144424, 6694736236093168200, 538028902298395832832, 46558260925421295229568, 4316186393637505403773328
Offset: 0
- H. W. Gould, personal communication.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
-
Table[Sum[(-1)^(n-k) * Binomial[n,k] * Binomial[(k+1)^2, n], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Dec 13 2020 *)
A003102
Largest number divisible by all numbers < its n-th root.
Original entry on oeis.org
2, 24, 420, 27720, 720720, 36756720, 5354228880, 481880599200, 25619985190800, 10685862914126400, 876240758958364800, 113035057905629059200, 24792356033967973651200, 9690712164777231700912800, 2364533768205644535022723200, 396059406174445459616306136000
Offset: 1
- A. Murthy, An application of Smarandache LCM sequence and the largest number divisible by all the integers not exceeding the r-th root, Preprint.
- N. Ozeki, On the problem 1, 2, 3, ..., [ n^(1/k) ] | n, Journal of the College of Arts and Sciences, Chiba University (Chiba, Japan), Vol. 3, No. 4 (Sept. 1962), pp. 427-431 [ Math. Rev. 30 213(1085) 1965 ].
- J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 277.
- D. O. Shklyarsky, N. N. Chentsov and I. M. Yaglom, Selected Problems and Theorems in Elementary Mathematics; Problem 78; Mir Publishers, Moscow.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- T. D. Noe, Table of n, a(n) for n = 1..50
- Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.
- A. Murthy, Some New Smarandache Sequences, Functions and Partitions, Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000, p. 179.
- N. Ozeki, On the problem 1, 2, 3, ..., [ n^(1/k) ] | n, Journal of the College of Arts and Sciences, Chiba University (Chiba, Japan), Vol. 3, No. 4 (Sept. 1962), pp. 427-431 [ Math. Rev. 30 213(1085) 1965 ]. [Annotated scanned copy]
- D. L. Silverman, Problem 159, Pi Mu Epsilon Journal, Vol. 4, No. 3, Fall 1965, p. 124.
- D. L. Silverman, Problem 159, Pi Mu Epsilon Journal, Vol. 4, No. 3, Fall 1965, p. 124. [Annotated scanned copy]
- Smarandache web site
-
k=1; lc=1; Table[While[r=Floor[lc^(1/n)]; Union[Mod[lc,Range[r]]]=={0}, k++; good=lc; lc=LCM[lc,k]]; m=2; While[r=Floor[(m*good)^(1/n)]; Union[Mod[m*good,Range[r]]]=={0}, m++ ]; m=m-1; m*good, {n,16}] (* T. D. Noe, Aug 01 2006 *)
Corrected and extended by
T. D. Noe, Aug 01 2006
A307093
a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k^2).
Original entry on oeis.org
1, 0, -1, -2, -2, 1, 10, 29, 63, 117, 191, 265, 264, -12, -1014, -3654, -9634, -21929, -45424, -87551, -158289, -267616, -415513, -563200, -561430, 12625, 2202084, 8368243, 23532027, 57848882, 131000395, 279675274, 569701663, 1114392742, 2099105261, 3805794420
Offset: 0
-
a[n_] := Sum[(-1)^k * Binomial[n, k^2], {k, 0, n}]; Array[a, 36, 0] (* Amiram Eldar, May 20 2021 *)
-
{a(n) = sum(k=0, sqrtint(n), (-1)^k*binomial(n, k^2))}
A177812
Triangular array read by rows: binomial(n,k^2), k=0..floor(sqrt(n)).
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 5, 5, 1, 6, 15, 1, 7, 35, 1, 8, 70, 1, 9, 126, 1, 1, 10, 210, 10, 1, 11, 330, 55, 1, 12, 495, 220, 1, 13, 715, 715, 1, 14, 1001, 2002, 1, 15, 1365, 5005, 1, 16, 1820, 11440, 1, 1, 17, 2380, 24310, 17, 1, 18, 3060, 48620, 153
Offset: 0
{1},
{1, 1},
{1, 2},
{1, 3},
{1, 4, 1},
{1, 5, 5},
{1, 6, 15},
{1, 7, 35},
{1, 8, 70},
{1, 9, 126, 1},
{1, 10, 210, 10}
-
Clear[t, n, m];
t[n_, m_] = Binomial[n, m^2];
Table[Table[t[n, m], {m, 0, Floor[Sqrt[n]]}], {n, 0, 10}];
Flatten[%]
-
tabf(nn) = {for (n = 0, nn, for (k = 0, sqrtint(n), print1(binomial(n, k^2), ", ");); print(););} \\ Michel Marcus, Feb 13 2014
Clarified definition, changed keyword to tabf. -
N. J. A. Sloane, Dec 16 2010
A177815
Triangle read by rows: binomial(n, m^3).
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 1, 9, 9, 1, 10, 45, 1, 11, 165, 1, 12, 495, 1, 13, 1287, 1, 14, 3003, 1, 15, 6435, 1, 16, 12870, 1, 17, 24310, 1, 18, 43758, 1, 19, 75582, 1, 20, 125970
Offset: 0
{1},
{1, 1},
{1, 2},
{1, 3},
{1, 4},
{1, 5},
{1, 6},
{1, 7},
{1, 8, 1},
{1, 9, 9},
{1, 10, 45},
{1, 11, 165},
{1, 12, 495},
{1, 13, 1287},
{1, 14, 3003},
{1, 15, 6435},
{1, 16, 12870},
{1, 17, 24310},
{1, 18, 43758},
{1, 19, 75582},
{1, 20, 125970},
...
{1, 27, 2220075, 1}
-
t[n_, m_] = Binomial[n, m^3];
Table[Table[t[n, m], {m, 0, Floor[n^(1/3)]}], {n, 0, 20}];
Flatten[%]
A177822
Sub-triangle of A008292: terms with square indices.
Original entry on oeis.org
1, 1, 1, 1, 4, 1, 11, 1, 26, 1, 1, 57, 57, 1, 120, 1191, 1, 247, 15619, 1, 502, 156190, 1, 1013, 1310354, 1, 1, 2036, 9738114, 2036
Offset: 0
{1},
{1, 1},
{1, 4},
{1, 11},
{1, 26, 1},
{1, 57, 57},
{1, 120, 1191},
{1, 247, 15619},
{1, 502, 156190},
{1, 1013, 1310354, 1},
{1, 2036, 9738114, 2036}
-
<< DiscreteMath`Combinatorica`
a = Table[Table[Eulerian[n + 1, m^2], {m, 0, Floor[Sqrt[n]]}], {n, 0, 10}];
Flatten[%]
A369406
a(n) = Sum_{k=0..n} binomial(n,k^3).
Original entry on oeis.org
1, 2, 3, 4, 5, 6, 7, 8, 10, 19, 56, 177, 508, 1301, 3018, 6451, 12887, 24328, 43777, 75602, 125991, 203512, 319793, 490338, 735496, 1081601, 1562302, 2220104, 3108162, 4292581, 5857016, 7920222, 10719709, 14991758, 23535855, 47071676, 124403657, 386938194, 1252225819
Offset: 0
-
Table[Sum[Binomial[n, k^3], {k, 0, n^(1/3)}], {n, 0, 38}]
nmax = 38; CoefficientList[Series[(1/(1 - x)) Sum[(x/(1 - x))^k^3, {k, 0, nmax}], {x, 0, nmax}], x]
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