A127935 Number of meaningful differential operations of the n-th order on the space R^(2+n).

3, 6, 16, 26, 84, 126, 424, 610, 2068, 2936, 9816, 13884, 45608, 64750, 208336, 297570, 938676, 1351492, 4181752, 6071028, 18454648, 27023598, 80796336, 119300636, 351331464, 522981328, 1518742384, 2278188504, 6531607248, 9869753934, 27963677600, 42547990626

Offset

1,1

References

R. Bott, L. W. Tu, Differential forms in algebraic topology, New York: Springer, 1982.

Links

Table of n, a(n) for n=1..32.

Branko Malesevic, Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33; arXiv:0704.0750 [math.DG], 2007.

Example

a(1) = 3 = A020701(1) is number of meaningful differential operations of the first order on the space R^3, namely {div, grad, curl}.
a(2) = 6 = A090989(2) is number of meaningful differential operations of the 2nd order on the space R^4 (some of them are identically zero though).
a(3) = 16 = A090990(3) is number of meaningful differential operations of the 3rd order on the space R^5.

Mathematica

r[n_] := Table[Boole[j == i + 1 || i + j == n + 1], {i, n}, {j, n}];
Table[Total@Total@If[n == 1, IdentityMatrix[3], MatrixPower[r[n+2], n-1]], {n, 10}]
(* Andrey Zabolotskiy, Apr 30 2021 *)

Crossrefs

Main diagonal of A116183.

Cf. A020701, A090989-A090995, A129638, A129639.

Sequence in context: A291803 A330017 A212977 * A144538 A130596 A280730

Adjacent sequences: A127932 A127933 A127934 * A127936 A127937 A127938

Keyword

nonn

Author

Jonathan Vos Post, Apr 09 2007, Jun 08 2007

Extensions

Corrected from 8th term onwards. It appears the 8th and 9th terms listed were incorrectly taken from A000045 with two numbers concatenated together, whereas the 8th, 9th and 10th terms should have been the 8th term of A090995, the 9th of A129638 and the 10th of A129639. Joseph Myers, Dec 23 2008

Name and examples corrected, terms a(11) and beyond added by Andrey Zabolotskiy, Apr 30 2021

Status

approved