A324354
Total number of occurrences of 4 in the (signed) displacement sets of all permutations of [n+4] divided by 4!.
Original entry on oeis.org
0, 1, 9, 76, 679, 6576, 69299, 792926, 9812079, 130741156, 1867777339, 28494131106, 462487232519, 7959671021576, 144813873037539, 2777366346993766, 56009230972732639, 1184896664408025036, 26240470547134420619, 607133649024919944266, 14649976322598313989879
Offset: 0
-
a:= n-> (k-> -add((-1)^j*binomial(n, j)*(n+k-j)!, j=1..n)/k!)(4):
seq(a(n), n=0..23);
-
m = 23;
CoefficientList[(1-Exp[-x])/(1-x)^5 + O[x]^(m+1), x]*Range[0, m]! (* Jean-François Alcover, May 03 2021 *)
A324355
Total number of occurrences of 5 in the (signed) displacement sets of all permutations of [n+5] divided by 5!.
Original entry on oeis.org
0, 1, 11, 109, 1115, 12151, 142205, 1788361, 24118967, 347811859, 5345895929, 87298986325, 1510075068419, 27590646911023, 531082929791861, 10743610293871681, 227906995674679535, 5059315590877577131, 117308151182930808977, 2835988521500605314829
Offset: 0
-
a:= n-> (k-> -add((-1)^j*binomial(n, j)*(n+k-j)!, j=1..n)/k!)(5):
seq(a(n), n=0..23);
-
m = 23;
CoefficientList[(1-Exp[-x])/(1-x)^6 + O[x]^(m+1), x]*Range[0, m]! (* Jean-François Alcover, May 03 2021 *)
A324356
Total number of occurrences of 6 in the (signed) displacement sets of all permutations of [n+6] divided by 6!.
Original entry on oeis.org
0, 1, 13, 148, 1707, 20686, 266321, 3652608, 53339831, 827870338, 13624599309, 237169578724, 4356110013107, 84220077081414, 1710164008931657, 36396070427846536, 810244122520224111, 18833465673721387018, 456310533309915775301, 11505888654389005045548
Offset: 0
-
a:= n-> (k-> -add((-1)^j*binomial(n, j)*(n+k-j)!, j=1..n)/k!)(6):
seq(a(n), n=0..23);
-
With[{nn=20},CoefficientList[Series[(1-Exp[-x])/(1-x)^7,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jul 06 2021 *)
A324357
Total number of occurrences of 7 in the (signed) displacement sets of all permutations of [n+7] divided by 7!.
Original entry on oeis.org
0, 1, 15, 193, 2479, 33081, 464807, 6906257, 108589887, 1805179321, 31676392519, 585609896433, 11383428770303, 232204651095353, 4961029124266599, 110811507291845521, 2583228239189752447, 62748345739947178617, 1585780756628964990407, 41635723030339339863281
Offset: 0
A324358
Total number of occurrences of 8 in the (signed) displacement sets of all permutations of [n+8] divided by 8!.
Original entry on oeis.org
0, 1, 17, 244, 3455, 50356, 766943, 12274858, 206788751, 3666278080, 68339173319, 1337340802942, 27431518405607, 588814390368244, 13204430589422015, 308877966133175746, 7525275697320564383, 190678032594396773128, 5017985343328106906711, 136977444553573371090790
Offset: 0
A324359
Total number of occurrences of 9 in the (signed) displacement sets of all permutations of [n+9] divided by 9!.
Original entry on oeis.org
0, 1, 19, 301, 4659, 73651, 1208849, 20736801, 372683159, 7020426511, 138543438429, 2861318625661, 61767341913539, 1391789835244251, 32689488282841529, 799220290375798681, 20312800343333343279, 535995638431063608871, 14665906835087251866389
Offset: 0
-
[0] cat [(-1/Factorial(9)) * &+[(-1)^j * Binomial(n,j) * Factorial(n+9-j) : j in [1..n]]: n in [1..20]]; // Vincenzo Librandi, Jun 06 2019
-
a:= n-> (k-> -add((-1)^j*binomial(n, j)*(n+k-j)!, j=1..n)/k!)(9):
seq(a(n), n=0..23);
-
Range[0, 20]! CoefficientList[Series[(1 - Exp[-x])/(1 - x)^10, {x, 0, 20}], x] (* Vincenzo Librandi, Jun 06 2019 *)
A324360
Total number of occurrences of 10 in the (signed) displacement sets of all permutations of [n+10] divided by 10!.
Original entry on oeis.org
0, 1, 21, 364, 6115, 104226, 1834205, 33576236, 641293047, 12792063934, 266464077769, 5792423481120, 131276423686979, 3098383343174978, 76066855087291221, 1940223116685166996, 51356370210296015215, 1409053932006095867526, 40028877611196977481857
Offset: 0
-
a:= n-> (k-> -add((-1)^j*binomial(n, j)*(n+k-j)!, j=1..n)/k!)(10):
seq(a(n), n=0..23);
-
With[{nn=20},CoefficientList[Series[(1-Exp[-x])/(1-x)^11,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 04 2023 *)