A027483
Second subdiagonal of triangle A027478, constructed from the Stirling numbers of the first kind.
Original entry on oeis.org
176, 2951, 22535, 112435, 426475, 1333906, 3614226, 8762370, 19439970, 40113425, 77924561, 143844701, 254168005, 432404980, 711642100, 1137438516, 1771335876, 2695062315, 4015516715, 5870624375, 8436160271, 11933641126, 16639392550, 22894902550, 31118577750
Offset: 3
A027489
First column of Triangle A027478, constructed from Stirling numbers of the first kind.
Original entry on oeis.org
1, 7, 176, 10746, 1297704, 272866980, 91570835040, 46034917019280, 33038572997888640, 32591683412232799680, 42861143959833044563200, 73273310483627217731644800, 159431451667479363623304936960
Offset: 1
Cf.
A027478 (Cube of normalized Stirling matrix of the first kind).
A027490
Second column of Triangle A027478, constructed from the Stirling numbers of the first kind.
Original entry on oeis.org
1, 39, 2951, 407450, 94128364, 33910601508, 18030534782364, 13546779499777104, 13886615636338251456, 18871622607827176957440, 33195072622146083265245760, 74062306101954993414777244800
Offset: 2
Cf.
A027478 (Cube of normalized Stirling matrix of the first kind).
A027491
Third column of Triangle A027478, constructed from the Stirling numbers of the first kind.
Original entry on oeis.org
1, 126, 22535, 6139575, 2487385684, 1446119232796, 1166500896454844, 1267622549125064100, 1809397702962603426816, 3319681647209765132077992, 7683477487469739839805775776, 22073571178683618465281583731376
Offset: 3
Cf.
A027478 (Cube of the normalized Stirling matrix of the first kind).
A027479
Triangle of the fourth power of the normalized, unsigned Stirling matrix of the first kind.
Original entry on oeis.org
1, 15, 1, 1175, 120, 1, 294330, 36935, 510, 1, 181082204, 25816200, 460035, 1560, 1, 231844265940, 36133755364, 757122975, 3411835, 3885, 1, 551220029003580, 91850446178400, 2159098539409, 11690792400, 18037810, 8400, 1, 2239429013789400720, 393327895035809180, 10088942720014620, 62324463343569, 117282133080, 75042450, 16380, 1
Offset: 1
First rows of the triangle are:
1;
15, 1;
1175, 120, 1;
294330, 36935, 510, 1;
181082204, 25816200, 460035, 1560, 1;
...
-
Module[{nmax=8,m},m=(Table[Table[(-1)^(i+j) StirlingS1[i,j]/i!,{j,1,nmax}],{i,1,nmax}]);m=m.m.m.m*Table[i!^4,{i,1,nmax}]; Flatten[Table[Table[m[[i,j]],{j,1,i}],{i,1,nmax}],1]]
Definition, formula and program edited for clarity by
Olivier Gérard, Jan 20 2019
A027477
Triangle of the square of the normalized, unsigned Stirling matrix of the first kind.
Original entry on oeis.org
1, 3, 1, 23, 12, 1, 330, 215, 30, 1, 7604, 5700, 1035, 60, 1, 256620, 212464, 45675, 3535, 105, 1, 11923260, 10645152, 2582209, 241080, 9730, 168, 1, 729524880, 691560092, 183962268, 19661649, 970200, 23058, 252, 1
Offset: 1
First rows of the triangle are:
1,
3,1,
23,12,1,
330,215,30,1,
7604,5700,1035,60,1,
256620,212464,45675,3535,105,1
...
-
Module[{nmax=8,m},m=(Table[Table[(-1)^(i+j) StirlingS1[i,j]/i!,{j,1,nmax}],{i,1,nmax}]);m=m.m*Table[i!^2,{i,1,nmax}]; Flatten[Table[Table[m[[i,j]],{j,1,i}],{i,1,nmax}],1]]
Definition, formula and program edited for clarity by
Olivier Gérard, Jan 20 2019
A027482
a(n) = n*(n^3 - 1)/2.
Original entry on oeis.org
7, 39, 126, 310, 645, 1197, 2044, 3276, 4995, 7315, 10362, 14274, 19201, 25305, 32760, 41752, 52479, 65151, 79990, 97230, 117117, 139909, 165876, 195300, 228475, 265707, 307314, 353626, 404985, 461745, 524272, 592944, 668151
Offset: 2
- Vincenzo Librandi, Table of n, a(n) for n = 2..1000
- S. Gartenhaus, Odd order pandiagonal latin and magic cubes in three and four dimensions, arXiv:math/0210275 [math.CO], 2002.
- Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
First subdiagonal of
A027478 (Cube of a triangular matrix constructed from the Stirling numbers of the first kind).
-
[n * (n^3 - 1)/2: n in [2..50]]; // Vincenzo Librandi, Dec 29 2012
-
Table[(m^4 - m)/2, {m, 44}] (* Zerinvary Lajos, Mar 21 2007 *)
CoefficientList[Series[(7 + 4*x + x^2)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 29 2012 *)
-
t(n)=n*(n+1)/2;
for(n=0,50,print1(t(n^2)-t(n)","))
Showing 1-7 of 7 results.
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