A115067 -id:A115067 - cp's OEIS frontend

A210638 Iterated Rényi numbers. Square array A(n,k), n>=0, k>=0, read by antidiagonals, A(n,k) = exponential transform applied n times to the constant function -1, evaluated at k.

-1, 1, -1, 1, -1, -1, 1, -1, 0, -1, 1, -1, 1, 1, -1, 1, -1, 2, 0, 1, -1, 1, -1, 3, -4, -2, -2, -1, 1, -1, 4, -11, 8, 2, -9, -1, 1, -1, 5, -21, 49, -14, 9, -9, -1, 1, -1, 6, -34, 139, -255, 13, -24, 50, -1, 1, -1, 7, -50, 296, -1106, 1508, 45, -80, 267, -1, 1

Offset: 0

Peter Luschny, Mar 26 2012

Motivation: The exponential transform applied n times to the constant function 1 evaluated at k was studied by E. T. Bell (Iterated Bell numbers, see A144150).

			n\k [0]  [1] [2]   [3]   [4]     [5]     [6]
[0] -1   -1  -1    -1    -1      -1      -1
[1]  1   -1   0     1     1      -2      -9  [A000587]
[2]  1   -1   1     0    -2       2       9
[3]  1   -1   2    -4     8     -14      13
[4]  1   -1   3   -11    49    -255    1508
[5]  1   -1   4   -21   139   -1106   10244
[6]  1   -1   5   -34   296   -3132   38916
column3(n) = (-2+7*n-3*n^2)/2  [A115067]
column4(n) = (-2+21*n-23*n^2+6*n^3)/2
column5(n) = (-6+199*n-405*n^2+245*n^3-45*n^4)/6
column6(n) = (-24+2866*n-9213*n^2+9470*n^3-3855*n^4+540*n^5 )/24

R. E. Beard, On the coefficients in the expansion of e^e^t and e^e^(-t), J. Inst. Actuar. 76 (1950), 152-163.
Alfréd Rényi, New methods and results in combinatorial analysis. (Paper is in Hungarian.) I. MTA III Oszt. Ivozl., 16 (1966), 77-105.

E. T. Bell, The iterated exponential integers, Ann. Math. 39(3) (1938), 539-557.
Antal E. Fekete, Apropos Bell and Stirling Numbers, Crux Mathematicorum with Mathematical Mayhem, Canadian Mathematical Society, Volume 25 Number 5 (May 1999), 274-281.
Peter Luschny, Set partitions and Bell numbers
V. R. Rao Uppuluri and J. A. Carpenter, Numbers generated by the function exp(1-e^x), Fib. Quart. 7 (1969), 437-448.

Cf. A000587, A144150, A209631.

exptr := proc(p) local g; g := proc(n) option remember; local k;
`if`(n=0, 1, add(binomial(n-1, k-1)*p(k)*g(n-k), k=1..n)) end end:
A210638 := (n, k) -> (exptr@@n)(-1)(k):
seq(lprint(seq(A210638(n, k), k=0..6)), n=0..6);

exptr[p_] := Module[{g}, g[n_] := g[n] = If[n==0, 1, Sum[Binomial[n-1, k-1] p[k] g[n-k], {k, 1, n}]]; g];
A[n_, k_] := Nest[exptr, -1&, n][k];
Table[A[n-k, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 29 2019 *)

A266387 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 322560.

Original entry on oeis.org

0, 0, 0, 0, 0, 7, 42, 147, 392, 882, 1764, 3234, 5544, 9009, 14014, 21021, 30576, 43316, 59976, 81396, 108528, 142443, 184338, 235543, 297528, 371910, 460460, 565110, 687960, 831285, 997542, 1189377, 1409632, 1661352, 1947792, 2272424, 2638944, 3051279

Offset: 1

Philippe A.J.G. Chevalier, Dec 28 2015

The sequence was discovered by enumerating all orbits of Aut(Z^7) and sorting the orbits as function of the infinity norm of the representative integer lattice points. This sequence is one of the 30 sequences that are obtained by classifying the orbits in a table with the rows being the infinity norm and the columns being the 30 cardinalities (1, 14, 84, 128, 168, 280, 448, 560, 672, 840, 896, 1680, 2240, 2688, 3360, 4480, 5376, 6720, 8960, 13440, 17920, 20160, 26880, 40320, 53760, 80640, 107520, 161280, 322560, 645120) generated by signed permutations of integer lattice points of Z^7.

The continued fraction expansion of this sequence is finite and simplifies to the g.f. 7*x^6/(1-x)^6 (see Mathematica). - Benedict W. J. Irwin, Feb 09 2016

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Other sequences that give the number of orbits of Aut(Z^7) as function of the infinity norm for different cardinalities of these orbits: A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A001477, A000217.

Join[{0, 0, 0, 0, 0},Table[Abs[SeriesCoefficient[Series[7/(x+6/(x - 5/2/(x + ContinuedFractionK[If[Mod[k, 2] ==0, (7 + k/2)/(6 + 2 k), ((k + 1)/2 - 5)/(2 (k - 1) +6)], x, {k, 0, 8}]))), {x, Infinity, 101}],2 n + 1]], {n, 0, 50}]] - (* Benedict W. J. Irwin, Feb 09 2016 *)

concat(vector(5), Vec(7*x^6/(1-x)^6 + O(x^50))) \\ Colin Barker, May 04 2016

From Colin Barker, Dec 29 2015: (Start)
a(n) = 7*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)/120.
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) for n>6.
G.f.: 7*x^6 / (1-x)^6.
(End)

A266395 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 161280.

Original entry on oeis.org

0, 0, 0, 0, 15, 75, 225, 525, 1050, 1890, 3150, 4950, 7425, 10725, 15015, 20475, 27300, 35700, 45900, 58140, 72675, 89775, 109725, 132825, 159390, 189750, 224250, 263250, 307125, 356265, 411075, 471975, 539400, 613800, 695640, 785400, 883575, 990675, 1107225

Offset: 1

Philippe A.J.G. Chevalier, Dec 29 2015

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A001477, A000217.

concat(vector(4), Vec(15*x^5/(1-x)^5 + O(x^50))) \\ Colin Barker, May 05 2016

From Colin Barker, Dec 29 2015: (Start)
a(n) = 5*(n-1)*(n-2)*(n-3)*(n-4)/8 = 15*A000332(n-1).
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>5.
G.f.: 15*x^5 / (1-x)^5.
(End)

A266396 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 80640.

Original entry on oeis.org

0, 0, 0, 10, 41, 105, 215, 385, 630, 966, 1410, 1980, 2695, 3575, 4641, 5915, 7420, 9180, 11220, 13566, 16245, 19285, 22715, 26565, 30866, 35650, 40950, 46800, 53235, 60291, 68005, 76415, 85560, 95480, 106216, 117810, 130305, 143745, 158175, 173641, 190190

Offset: 1

Philippe A.J.G. Chevalier, Dec 29 2015

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A001477, A000217.

LinearRecurrence[{5,-10,10,-5,1},{0,0,0,10,41},50] (* Harvey P. Dale, Nov 18 2024 *)

concat(vector(3), Vec(x^4*(10-9*x)/(1-x)^5 + O(x^50))) \\ Colin Barker, May 05 2016

From Colin Barker, Dec 29 2015: (Start)
a(n) = (n^4+30*n^3-205*n^2+390*n-216)/24.
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>5.
G.f.: x^4*(10-9*x) / (1-x)^5.
(End)

A266397 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 26880.

Original entry on oeis.org

0, 0, 9, 31, 70, 130, 215, 329, 476, 660, 885, 1155, 1474, 1846, 2275, 2765, 3320, 3944, 4641, 5415, 6270, 7210, 8239, 9361, 10580, 11900, 13325, 14859, 16506, 18270, 20155, 22165, 24304, 26576, 28985, 31535, 34230, 37074, 40071, 43225, 46540, 50020, 53669

Offset: 1

Philippe A.J.G. Chevalier, Dec 29 2015

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A001477, A000217.

concat(vector(2), Vec(x^3*(9-5*x)/(1-x)^4 + O(x^50))) \\ Colin Barker, May 05 2016

From Colin Barker, Dec 29 2015: (Start)
a(n) = (4*n^3+3*n^2-37*n+30)/6.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4.
G.f.: x^3*(9-5*x) / (1-x)^4.
(End)

A266398 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 13440.

Original entry on oeis.org

0, 0, 12, 37, 76, 130, 200, 287, 392, 516, 660, 825, 1012, 1222, 1456, 1715, 2000, 2312, 2652, 3021, 3420, 3850, 4312, 4807, 5336, 5900, 6500, 7137, 7812, 8526, 9280, 10075, 10912, 11792, 12716, 13685, 14700, 15762, 16872, 18031, 19240, 20500, 21812, 23177

Offset: 1

Philippe A.J.G. Chevalier, Dec 29 2015

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002112, A045943, A115067, A008586, A008585, A005843, A001477, A000217.

concat(vector(2), Vec(x^3*(12-11*x)/(1-x)^4 + O(x^50))) \\ Colin Barker, May 05 2016

From Colin Barker, Dec 29 2015: (Start)
a(n) = (n^3+30*n^2-97*n+66)/6.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4.
G.f.: x^3*(12-11*x) / (1-x)^4.
(End)

A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 6, 7, 7, 1, 5, 8, 10, 11, 11, 1, 6, 10, 13, 15, 16, 16, 1, 7, 12, 16, 19, 21, 22, 22, 1, 8, 14, 19, 23, 26, 28, 29, 29, 1, 9, 16, 22, 27, 31, 34, 36, 37, 37, 1, 10, 18, 25, 31, 36, 40, 43, 45, 46, 46, 1, 11, 20, 28, 35, 41

Offset: 0

Franck Maminirina Ramaharo, Apr 20 2018

Columns are linear recurrence sequences with signature (3,-3,1).
8*T(n,k) + A166147(k-1) are squares.
Columns k are binomial transforms of [1, k, 1, 0, 0, 0, ...].
Antidiagonals sums yield A116731.

			The array T(n,k) begins
1    1    1    1    1    1    1    1    1    1    1    1    1  ...  A000012
1    2    3    4    5    6    7    8    9   10   11   12   13  ...  A000027
2    4    6    8   10   12   14   16   18   20   22   24   26  ...  A005843
4    7   10   13   16   19   22   25   28   31   34   37   40  ...  A016777
7   11   15   19   23   27   31   35   39   43   47   51   55  ...  A004767
11  16   21   26   31   36   41   46   51   56   61   66   71  ...  A016861
16  22   28   34   40   46   52   58   64   70   76   82   88  ...  A016957
22  29   36   43   50   57   64   71   78   85   92   99  106  ...  A016993
29  37   45   53   61   69   77   85   93  101  109  117  125  ...  A004770
37  46   55   64   73   82   91  100  109  118  127  136  145  ...  A017173
46  56   66   76   86   96  106  116  126  136  146  156  166  ...  A017341
56  67   78   89  100  111  122  133  144  155  166  177  188  ...  A017401
67  79   91  103  115  127  139  151  163  175  187  199  211  ...  A017605
79  92  105  118  131  144  157  170  183  196  209  222  235  ...  A190991
...
The inverse binomial transforms of the columns are
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    1    2    3    4    5    6    7    8    9   10   11   12  ...
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
...
T(k,n-k) = A087401(n,k) + 1 as triangle
1
1   1
1   2   2
1   3   4   4
1   4   6   7   7
1   5   8  10  11  11
1   6  10  13  15  16  16
1   7  12  16  19  21  22  22
1   8  14  19  23  26  28  29  29
1   9  16  22  27  31  34  36  37  37
1  10  18  25  31  36  40  43  45  46  46
...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

Cf. A085475, A086270, A086271, A086272, A086273, A130154, A159798, A162609, A162610, A300401.

T := (n, k) -> binomial(n, 2) + k*n + 1;
for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;

Table[With[{n = m - k}, Binomial[n, 2] + k n + 1], {m, 0, 11}, {k, m, 0, -1}] // Flatten (* Michael De Vlieger, Apr 21 2018 *)

T(n, k) := binomial(n, 2)+ k*n + 1$
for n:0 thru 20 do
    print(makelist(T(n, k), k, 0, 20));

T(n,k) = binomial(n, 2) + k*n + 1;
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 17 2018

G.f.: (3*x^2*y - 3*x*y + y - 2*x^2 + 2*x - 1)/((x - 1)^3*(y - 1)^2).
E.g.f.: (1/2)*(2*x*y + x^2 + 2)*exp(y + x).
T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k), with T(0,k) = 1, T(1,k) = k + 1 and T(2,k) = 2*k + 2.
T(n,k) = T(n-1,k) + n + k - 1.
T(n,k) = T(n,k-1) + n, with T(n,0) = 1.
T(n,0) = A152947(n+1).
T(n,1) = A000124(n).
T(n,2) = A000217(n).
T(n,3) = A034856(n+1).
T(n,4) = A052905(n).
T(n,5) = A051936(n+4).
T(n,6) = A246172(n+1).
T(n,7) = A302537(n).
T(n,8) = A056121(n+1) + 1.
T(n,9) = A056126(n+1) + 1.
T(n,10) = A051942(n+10) + 1, n > 0.
T(n,11) = A101859(n) + 1.
T(n,12) = A132754(n+1) + 1.
T(n,13) = A132755(n+1) + 1.
T(n,14) = A132756(n+1) + 1.
T(n,15) = A132757(n+1) + 1.
T(n,16) = A132758(n+1) + 1.
T(n,17) = A212427(n+1) + 1.
T(n,18) = A212428(n+1) + 1.
T(n,n) = A143689(n) = A300192(n,2).
T(n,n+1) = A104249(n).
T(n,n+2) = T(n+1,n) = A005448(n+1).
T(n,n+3) = A000326(n+1).
T(n,n+4) = A095794(n+1).
T(n,n+5) = A133694(n+1).
T(n+2,n) = A005449(n+1).
T(n+3,n) = A115067(n+2).
T(n+4,n) = A133694(n+2).
T(2*n,n) = A054556(n+1).
T(2*n,n+1) = A054567(n+1).
T(2*n,n+2) = A033951(n).
T(2*n,n+3) = A001107(n+1).
T(2*n,n+4) = A186353(4*n+1) (conjectured).
T(2*n,n+5) = A184103(8*n+1) (conjectured).
T(2*n,n+6) = A250657(n-1) = A250656(3,n-1), n > 1.
T(n,2*n) = A140066(n+1).
T(n+1,2*n) = A005891(n).
T(n+2,2*n) = A249013(5*n+4) (conjectured).
T(n+3,2*n) = A186384(5*n+3) = A186386(5*n+3) (conjectured).
T(2*n,2*n) = A143689(2*n).
T(2*n+1,2*n+1) = A143689(2*n+1) (= A030503(3*n+3) (conjectured)).
T(2*n,2*n+1) = A104249(2*n) = A093918(2*n+2) = A131355(4*n+1) (= A030503(3*n+5) (conjectured)).
T(2*n+1,2*n) = A085473(n).
a(n+1,5*n+1)=A051865(n+1) + 1.
a(n,2*n+1) = A116668(n).
a(2*n+1,n) = A054569(n+1).
T(3*n,n) = A025742(3*n-1), n > 1 (conjectured).
T(n,3*n) = A140063(n+1).
T(n+1,3*n) = A069099(n+1).
T(n,4*n) = A276819(n).
T(4*n,n) = A154106(n-1), n > 0.
T(2^n,2) = A028401(n+2).
T(1,n)*T(n,1) = A006000(n).
T(n*(n+1),n) = A211905(n+1), n > 0 (conjectured).
T(n*(n+1)+1,n) = A294259(n+1).
T(n,n^2+1) = A081423(n).
T(n,A000217(n)) = A158842(n), n > 0.
T(n,A152947(n+1)) = A060354(n+1).
floor(T(n,n/2)) = A267682(n) (conjectured).
floor(T(n,n/3)) = A025742(n-1), n > 0 (conjectured).
floor(T(n,n/4)) = A263807(n-1), n > 0 (conjectured).
ceiling(T(n,2^n)/n) = A134522(n), n > 0 (conjectured).
ceiling(T(n,n/2+n)/n) = A051755(n+1) (conjectured).
floor(T(n,n)/n) = A133223(n), n > 0 (conjectured).
ceiling(T(n,n)/n) = A007494(n), n > 0.
ceiling(T(n,n^2)/n) = A171769(n), n > 0.
ceiling(T(2*n,n^2)/n) = A046092(n), n > 0.
ceiling(T(2*n,2^n)/n) = A131520(n+2), n > 0.

cp's OEIS Frontend

A210638 Iterated Rényi numbers. Square array A(n,k), n>=0, k>=0, read by antidiagonals, A(n,k) = exponential transform applied n times to the constant function -1, evaluated at k.

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A266387 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 322560.

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A266395 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 161280.

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A266396 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 80640.

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A266397 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 26880.

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A266398 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 13440.

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A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

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