A045648 -id:A045648 - cp's OEIS frontend

A049075 Eigensequence of a power series transformation.

1, 1, 2, 4, 8, 18, 43, 102, 247, 617, 1564, 4003, 10355, 27051, 71225, 188743, 503111, 1348301, 3630294, 9815159, 26637436, 72540432, 198162708, 542875096, 1491126550, 4105602719, 11329408543, 31328137525, 86795258650, 240898943969, 669730499207, 1864855943748

Offset: 1

Michael Somos, Aug 08 1999

Euler transform of a(n) - if( n%4, 0, a(n/2)) is sequence itself with offset 0.

			x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 18*x^6 + 43*x^7 + 102*x^8 + 247*x^9 + 617*x^10 + ...

Alois P. Heinz, Table of n, a(n) for n = 1..650

Cf. A045648, A000081, A004111.

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1 else (add(d*p(d), d=divisors(n)) +add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n-1))/n fi end end: b:= etr(n-> a(n) -`if`(modp(n,4)<>0, 0,a(n/2))): a:= n-> b(n-1): seq(a(n), n=1..40);  # Alois P. Heinz, Sep 06 2008

s[ n_, k_ ] := s[ n, k ]=a[ n+1-k ]+If[ n<2k, 0, -s[ n-k, k ](-1)^k ]; a[ 1 ]=1; a[ n_ ] := a[ n ]=Sum[ a[ i ]s[ n-1, i ]i, {i, 1, n-1} ]/(n-1); Table[ a[ i ], {i, 1, 30} ]

{a(n) = local(A=x); if( n<1, 0, for( k=1, n-1, A *= (1 + (-x)^k + x*O(x^n))^((-1)^k * polcoeff(A, k))); polcoeff(A, n))}

G.f.: A(x) = x exp(A(x) - A(-x^2)/2 + A(x^3)/3 - A(-x^4)/4 + ...). Also A(x) = Sum_{n >= 1} a(n)*x^n = x * Product_{n >= 1} (1+(-x)^n)^((-1)^n*a(n)).
G.f.: x prod_{n>0} (1-x^(4n))^a(2n)/(1-x^n)^a(n).
a(n) ~ c * d^n / n^(3/2), where d = 2.92045137601697174071599643..., c = 0.4299447159290328896620383... . - Vaclav Kotesovec, Aug 25 2014

A355049 Number of chiral pairs of orthoplex n-ominoes with cell centers determining n-3 space.

Original entry on oeis.org

8, 76, 440, 2019, 8147, 30367, 107061, 361655, 1181761, 3762817, 11733393, 35957132, 108591703, 323914688, 955984083, 2795513143, 8108894051, 23354358683, 66838785954, 190211189706, 538567451991, 1517943035326

Offset: 7

Robert A. Russell, Jun 16 2022

Orthoplex polyominoes are connected sets of cells of regular tilings with Schläfli symbols {}, {4}, {3,4}, {3,3,4}, {3,3,3,4}, etc. These are tilings of regular orthoplexes projected on their circumspheres. Orthoplex polyominoes are equivalent to multidimensional polyominoes that do not extend more than two units along any axis, i.e., fit within a 2^d cube. Each member of a chiral pair is a reflection but not a rotation of the other.

			a(7)=8 because there are 8 pairs of chiral heptominoes in 2^4 space. See trunks 1, 6, 8, 12, 13, 19, 27, and 28 in the linked Trunk Generating Functions.

Robert A. Russell, Table of n, a(n) for n = 7..100
Robert A. Russell, Trunk Generating Functions

Cf. A355047 (oriented), A355048 (unoriented), A355050 (achiral) A355051 (asymmetric), A045648 (rooted chiral).

Other dimensions: A036368 (n-2), A045649 (n-1), A355054 (multidimensional).

sc[n_,k_] := sc[n,k] = c[n+1-k,1] + If[n<2k, 0, sc[n-k,k](-1)^k];
c[1,1] := 1; c[n_,1] := c[n,1] = Sum[c[i,1] sc[n-1,i]i, {i,1,n-1}]/(n-1);
c[n_,k_] := c[n, k] = Sum[c[i, 1] c[n-i, k-1], {i,1,n-1}];
nmax = 30; K[x_] := Sum[c[i,1] x^i, {i,0,nmax}]
Drop[CoefficientList[Series[(14 K[x]^6 + 3 K[x]^7 + 6 K[x]^4 K[-x^2] + 6 K[x]^5 K[-x^2] - 18 K[x]^2 K[-x^2]^2 + 3 K[x]^3 K[-x^2]^2 - 10 K[-x^2]^3 - 6 K[x] K[-x^2]^3 + 4 K[x^3]^2 - 6 K[x] K[-x^2] K[-x^4] + 4 K[-x^6]) / 24 + K[x]^3 (38 K[x]^4 + 9 K[x]^5 + 4 K[x]^2 K[-x^2] + 10 K[x]^3 K[-x^2] - 2 K[-x^2]^2 + K[x] K[-x^2]^2) / (8(1-K[x])) + K[x]^6 (5 K[x] + 16 K[x]^2 + 6 K[x]^3 + K[-x^2] + 2 K[x] K[-x^2]) / (2(1-K[x])^2) - K[-x^2]^2 (K[x]^4 + 2 K[x] K[-x^2] + 4 K[x]^2 K[-x^2] + 2 K[-x^2]^2 + 5 K[x] K[-x^2]^2 + K[-x^4] + K[x] K[-x^4]) / (4(1-K[-x^2])) + K[x]^7 (2 + 42 K[x] + 51 K[x]^2 + 24 K[x]^3 + 3 K[-x^2]) / (12(1-K[x])^3) + (K[x] K[x^3]^2) / (3(1-K[x^3])) - K[x]^2 K[-x^2]^2 (2 K[x] + 5 K[x]^3 + 2 K[-x^2] + K[x] K[-x^2]) / (4(1-K[x]) (1-K[-x^2])) - K[-x^2]^4 (8 + K[x] + 8 K[x] K[-x^2]) / (8(1-K[-x^2])^2) + K[x]^9 (17 + 8 K[x]) / (8(1-K[x])^4) - K[x]^5 (1 + 4 K[x]) K[-x^2]^2 / (4(1-K[x])^2 (1-K[-x^2])) + (K[x] K[-x^4]^2) / (4(1-K[-x^4])) + (3 K[x]^10) / (8(1-K[x])^5) - ((K[x]^6 K[-x^2]^2) / (4(1-K[x])^3 (1-K[-x^2]))) - (((1 + K[x]) K[-x^2]^5) / (4(1-K[-x^2])^3)) + ((1 + K[x]) K[-x^2] K[-x^4]^2) / (4(1-K[-x^2]) (1-K[-x^4])) - ((K[x]^2 K[-x^2]^4) / (8(1-K[x]) (1-K[-x^2])^2)), {x,0,nmax}], x], 7]

a(n) = A355047(n) - A355048(n) = (A355047(n) - A355050(n)) / 2 = A355048(n) - A355050(n).

G.f.: (14 C(x)^6 + 3 C(x)^7 + 6 C(x)^4 C(-x^2) + 6 C(x)^5 C(-x^2) - 18 C(x)^2 C(-x^2)^2 + 3 C(x)^3 C(-x^2)^2 - 10 C(-x^2)^3 - 6 C(x) C(-x^2)^3 + 4 C(x^3)^2 - 6 C(x) C(-x^2) C(-x^4) + 4 C(-x^6)) / 24 + C(x)^3 (38 C(x)^4 + 9 C(x)^5 + 4 C(x)^2 C(-x^2) + 10 C(x)^3 C(-x^2) - 2 C(-x^2)^2 + C(x) C(-x^2)^2) / (8(1-C(x))) + C(x)^6 (5 C(x) + 16 C(x)^2 + 6 C(x)^3 + C(-x^2) + 2 C(x) C(-x^2)) / (2(1-C(x))^2) - C(-x^2)^2 (C(x)^4 + 2 C(x) C(-x^2) + 4 C(x)^2 C(-x^2) + 2 C(-x^2)^2 + 5 C(x) C(-x^2)^2 + C(-x^4) + C(x) C(-x^4)) / (4(1-C(-x^2))) + C(x)^7 (2 + 42 C(x) + 51 C(x)^2 + 24 C(x)^3 + 3 C(-x^2)) / (12(1-C(x))^3) + (C(x) C(x^3)^2) / (3(1-C(x^3))) - C(x)^2 C(-x^2)^2 (2 C(x) + 5 C(x)^3 + 2 C(-x^2) + C(x) C(-x^2)) / (4(1-C(x)) (1-C(-x^2))) - C(-x^2)^4 (8 + C(x) + 8 C(x) C(-x^2)) / (8(1-C(-x^2))^2) + C(x)^9 (17 + 8 C(x)) / (8(1-C(x))^4) - C(x)^5 (1 + 4 C(x)) C(-x^2)^2 / (4(1-C(x))^2 (1-C(-x^2))) + (C(x) C(-x^4)^2) / (4(1-C(-x^4))) + (3 C(x)^10) / (8(1-C(x))^5) - ((C(x)^6 C(-x^2)^2) / (4(1-C(x))^3 (1-C(-x^2)))) - (((1 + C(x)) C(-x^2)^5) / (4(1-C(-x^2))^3)) + ((1 + C(x)) C(-x^2) C(-x^4)^2) / (4(1-C(-x^2)) (1-C(-x^4))) - ((C(x)^2 C(-x^2)^4) / (8(1-C(x)) (1-C(-x^2))^2)) where C(x) is the generating function for chiral n-ominoes in n-1 space, one cell labeled in A045648.

A045649 Number of chiral n-ominoes in n-1 space.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 5, 9, 15, 31, 70, 146, 300, 656, 1471, 3258, 7245, 16400, 37461, 85773, 197365, 457297, 1065070, 2489750, 5842741, 13766775, 32552087, 77208432, 183670145, 438176307, 1048092760, 2513081101

Offset: 1

Robert A. Russell

Lunnon's DR(n,n-1)-DE(n,n-1). Knuth describes methodology for a similar enumeration, that of free trees with n nodes.

D. E. Knuth, Fundamental Algorithms, 3d Ed. 1997, pp. 386-88.

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
W. F. Lunnon, Counting Multidimensional Polyominoes, Computer Journal, Vol. 18 (1975), pp. 366-67.

Cf. A045648, A000081, A004111.

s[ n_, k_ ] := s[ n, k ]=c[ n+1-k ]+If[ n<2k, 0, s[ n-k, k ](-1)^k ]; c[ 1 ]=1; c[ n_ ] := c[ n ]=Sum[ c[ i ]s[ n-1, i ]i, {i, 1, n-1} ]/(n-1); Table[ c[ i ]-Sum[ c[ j ]c[ i-j ], {j, 1, i/2} ]+If[ OddQ[ i ], 0, c[ i/2 ](c[ i/2 ]+(-1)^(i/2))/2 ], {i, 1, 33} ]

G.f.: C(x)-C^2(x)/2+C(-x^2)/2 where C(x) is g.f. for same sequence with one cell labeled, A045648.

a(n) ~ c * d^n / n^(5/2), where d = 2.58968405406171542574769690513208346256... and c = 0.36257350770010314582973624284... . - Vaclav Kotesovec, Feb 29 2016

A355054 Number of chiral pairs of multidimensional n-ominoes with cell centers determining n-3 space.

Original entry on oeis.org

6, 54, 297, 1341, 5468, 20519, 72168, 242886, 791780, 2514453, 7814225, 23863941, 71835845, 213601046, 628450974, 1832227629, 5299559865, 15221688836, 43450246045, 123345029035, 348417524877, 979803281560

Offset: 5

Robert A. Russell, Jun 16 2022

Multidimensional polyominoes are connected sets of cells of regular tilings with Schläfli symbols {oo}, {4,4}, {4,3,4}, {4,3,3,4}, etc. Each tile is a regular orthotope (hypercube). Each member of a chiral pair is a reflection but not a rotation of the other.

			a(5)=6 because there are 6 chiral pairs of pentominoes in 2-space.

Robert A. Russell, Table of n, a(n) for n = 5..100
W. F. Lunnon, Counting multidimensional polyominoes. Computer Journal 18 (1975), no. 4, pp. 366-367.
Robert A. Russell, Trunk Generating Functions

Cf. A355052 (oriented), A355053 (unoriented), A355055 (achiral) A355056 (asymmetric), A191092 (fixed), A045648 (rooted chiral), A195738 (Lunnon's DR), A049430 (Lunnon's DE).

Other dimensions: A036365 (n-2), A045649 (n-1), A355049 (orthoplex).

sc[n_,k_] := sc[n,k] = c[n+1-k,1] + If[n<2k, 0, sc[n-k,k](-1)^k];
c[1,1] := 1; c[n_,1] := c[n,1] = Sum[c[i,1] sc[n-1,i]i, {i,1,n-1}]/(n-1);
c[n_,k_] := c[n, k] = Sum[c[i, 1] c[n-i, k-1], {i,1,n-1}];
nmax = 30; K[x_] := Sum[c[i,1] x^i, {i,0,nmax}]
Drop[CoefficientList[Series[(12 K[x]^4 + 87 K[x]^5 + 50 K[x]^6 + 3 K[x]^7 + 18 K[x]^3 K[-x^2] + 36 K[x]^4 K[-x^2] + 6 K[x]^5 K[-x^2] - 12 K[-x^2]^2 - 27 K[x] K[-x^2]^2 - 6 K[x]^2 K[-x^2]^2 + 3 K[x]^3 K[-x^2]^2 - 16 K[-x^2]^3 - 6 K[x] K[-x^2]^3 + 4 K[x^3]^2 - 6 K[x] K[-x^4] - 6 K[x] K[-x^2] K[-x^4] + 4 K[-x^6]) / 24 + K[x]^2 (16 K[x]^3 + 159 K[x]^4 + 112 K[x]^5 + 9 K[x]^6 + 14 K[x]^2 K[-x^2] + 32 K[x]^3 K[-x^2] + 10 K[x]^4 K[-x^2] - K[-x^2]^2 + K[x]^2 K[-x^2]^2) / (8 (1-K[x])) + K[x]^5 (2 K[x] + 67 K[x]^2 + 46 K[x]^3 + 6 K[x]^4 + 3 K[-x^2] + 6 K[x] K[-x^2] + 2 K[x]^2 K[-x^2]) / (2 (1-K[x])^2) - K[-x^2] (2 K[x]^2 K[-x^2] + 7 K[-x^2]^2 + 17 K[x] K[-x^2]^2 + 2 K[x]^2 K[-x^2]^2 + 7 K[-x^2]^3 + 5 K[x] K[-x^2]^3 + K[-x^4] + K[x] K[-x^4] + K[-x^2] K[-x^4] + K[x] K[-x^2] K[-x^4]) / (4 (1-K[-x^2])) + K[x]^6 (4 K[x] + 153 K[x]^2 + 75 K[x]^3 + 12 K[x]^4 + 3 K[-x^2] + 3 K[x] K[-x^2]) / (6 (1-K[x])^3) - K[x]^2 K[-x^2]^2 (K[x] + K[-x^2]) / ((1-K[x]) (1-K[-x^2])) + (K[x] K[x^3]^2) / (3 (1-K[x^3])) + K[x]^9 (21 + 4 K[x]) / (2 (1-K[x])^4) - K[-x^2]^4 (6 + 7 K[x] + 2 K[-x^2] + 2 K[x] K[-x^2]) / (2 (1-K[-x^2])^2) + 3 K[x]^10 / (2 (1-K[x])^5) - K[x]^2 K[-x^2]^4 / (2 (1-K[x]) (1-K[-x^2])^2) - (1 + K[x]) K[-x^2]^5 / (1-K[-x^2])^3, {x,0,nmax}], x], 5]

a(n) = A355052(n) - A355053(n) = (A355052(n) - A355055(n)) / 2 = A355053(n) - A355055(n).
a(n) = A195738(n,n-3) - A049430(n,n-3), diagonals of Lunnon's DR and DE arrays.
G.f.: (12 C(x)^4 + 87 C(x)^5 + 50 C(x)^6 + 3 C(x)^7 + 18 C(x)^3 C(-x^2) + 36 C(x)^4 C(-x^2) + 6 C(x)^5 C(-x^2) - 12 C(-x^2)^2 - 27 C(x) C(-x^2)^2 - 6 C(x)^2 C(-x^2)^2 + 3 C(x)^3 C(-x^2)^2 - 16 C(-x^2)^3 - 6 C(x) C(-x^2)^3 + 4 C(x^3)^2 - 6 C(x) C(-x^4) - 6 C(x) C(-x^2) C(-x^4) + 4 C(-x^6)) / 24 + C(x)^2 (16 C(x)^3 + 159 C(x)^4 + 112 C(x)^5 + 9 C(x)^6 + 14 C(x)^2 C(-x^2) + 32 C(x)^3 C(-x^2) + 10 C(x)^4 C(-x^2) - C(-x^2)^2 + C(x)^2 C(-x^2)^2) / (8 (1-C(x))) + C(x)^5 (2 C(x) + 67 C(x)^2 + 46 C(x)^3 + 6 C(x)^4 + 3 C(-x^2) + 6 C(x) C(-x^2) + 2 C(x)^2 C(-x^2)) / (2 (1-C(x))^2) - C(-x^2) (2 C(x)^2 C(-x^2) + 7 C(-x^2)^2 + 17 C(x) C(-x^2)^2 + 2 C(x)^2 C(-x^2)^2 + 7 C(-x^2)^3 + 5 C(x) C(-x^2)^3 + C(-x^4) + C(x) C(-x^4) + C(-x^2) C(-x^4) + C(x) C(-x^2) C(-x^4)) / (4 (1-C(-x^2))) + C(x)^6 (4 C(x) + 153 C(x)^2 + 75 C(x)^3 + 12 C(x)^4 + 3 C(-x^2) + 3 C(x) C(-x^2)) / (6 (1-C(x))^3) - C(x)^2 C(-x^2)^2 (C(x) + C(-x^2)) / ((1-C(x)) (1-C(-x^2))) + (C(x) C(x^3)^2) / (3 (1-C(x^3))) + C(x)^9 (21 + 4 C(x)) / (2 (1-C(x))^4) - C(-x^2)^4 (6 + 7 C(x) + 2 C(-x^2) + 2 C(x) C(-x^2)) / (2 (1-C(-x^2))^2) + 3 C(x)^10 / (2 (1-C(x))^5) - C(x)^2 C(-x^2)^4 / (2 (1-C(x)) (1-C(-x^2))^2) - (1 + C(x)) C(-x^2)^5 / (1-C(-x^2))^3 where C(x) is the generating function for chiral n-ominoes in n-1 space, one cell labeled in A045648.

A036365 Number of chiral n-ominoes in n-2 space.

Original entry on oeis.org

0, 2, 6, 17, 49, 135, 361, 951, 2493, 6497, 16837, 43498, 112164, 288741, 742294, 1906552, 4893835, 12555662, 32201344, 82566738, 211675672, 542621858, 1390929877, 3565435302, 9139718572, 23430209922, 60069035611, 154014868677

Offset: 3

Robert A. Russell

a(n) is Lunnon's DR(n,n-2) - DE(n,n-2).

			0 chiral trominoes in 1-space;
2 pairs of chiral tetrominoes (L,S) in 2-space;
6 pairs of chiral pentominoes in 3-space.

W. F. Lunnon, Counting Multidimensional Polyominoes, Computer Journal, Vol. 18 (1975), pp. 366-67.

Cf. A045648, A045649, A036364.

sc[ n_, k_ ] := sc[ n, k ]=c[ n+1-k, 1 ]+If[ n<2k, 0, sc[ n-k, k ](-1)^k ]; c[ 1, 1 ] := 1;
c[ n_, 1 ] := c[ n, 1 ]=Sum[ c[ i, 1 ]sc[ n-1, i ]i, {i, 1, n-1} ]/(n-1);
c[ n_, k_ ] := c[ n, k ]=Sum[ c[ i, 1 ]c[ n-i, k-1 ], {i, 1, n-1} ];
Table[ c[ i, 3 ]/2+5c[ i, 4 ]/8+Sum[ c[ i, j ], {j, 5, i} ]+If[ OddQ[ i ], 0,
3c[ i/2, 2 ](-1)^(i/2)/8-If[ OddQ[ i/2 ], 0, c[ i/4, 1 ](-1)^(i/4)/4 ] ]
+Sum[ c[ j, 1 ](c[ i-2j, 1 ]/2+c[ i-2j, 2 ]/4)(-1)^j, {j, 1, (i-1)/2} ], {i, 3, 30} ]

G.f.: C^3(x)/2 + C(x)C(-x^2)/2 + 5C^4(x)/8 + C^2(x)C(-x^2)/4 + 3C^2(-x^2)/8 - C(-x^4)/4 + C^5(x)/(1-C(x)), where C(x) is the generating function for chiral n-ominoes in n-1 space, one cell labeled (that is, C(x) is the g.f. of the sequence A045648).

A306768 G.f. A(x) satisfies: A(x) = x*exp(-A(-x) + A(-x^2)/2 - A(-x^3)/3 + A(-x^4)/4 - A(-x^5)/5 + ...).

Original entry on oeis.org

0, 1, 1, -1, -2, 2, 6, -5, -18, 15, 59, -54, -215, 199, 813, -744, -3135, 2890, 12394, -11538, -50017, 46806, 204893, -192451, -849681, 800974, 3560927, -3367656, -15058478, 14279426, 64171736, -60992032, -275304665, 262199050, 1188070488, -1133572891, -5153913606

Offset: 0

Ilya Gutkovskiy, Apr 14 2019

sign

			G.f.: A(x) = x + x^2 - x^3 - 2*x^4 + 2*x^5 + 6*x^6 - 5*x^7 - 18*x^8 + 15*x^9 + 59*x^10 - 54*x^11 - 215*x^12 + ...

Cf. A000081, A004111, A045648, A049075, A073075, A115593, A307365, A307366, A307538.

terms = 36; A[] = 0; Do[A[x] = x Exp[Sum[(-1)^k A[-x^k]/k, {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = SeriesCoefficient[x Product[1/(1 + x^k)^((-1)^k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = 0; Table[a[n], {n, 0, 36}]

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} 1/(1 + x^n)^((-1)^n*a(n)).

Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+d)*d*a(d) ) * a(n-k+1).

A307365 G.f. A(x) satisfies: A(x) = x*exp(A(-x) + A(-x^2)/2 + A(-x^3)/3 + A(-x^4)/4 + ...).

Original entry on oeis.org

0, 1, -1, -1, 2, 1, -4, -3, 11, 10, -36, -32, 122, 105, -420, -368, 1497, 1336, -5491, -4919, 20477, 18393, -77397, -69883, 296306, 268711, -1146538, -1042924, 4475265, 4081598, -17600475, -16091719, 69681964, 63845971, -277494594, -254730047, 1110782803, 1021361912

Offset: 0

Ilya Gutkovskiy, Apr 05 2019

sign

			G.f.: A(x) = x - x^2 - x^3 + 2*x^4 + x^5 - 4*x^6 - 3*x^7 + 11*x^8 + 10*x^9 - 36*x^10 - 32*x^11 + ...

Cf. A000081, A004111, A045648, A049075, A307366.

terms = 37; A[] = 0; Do[A[x] = x Exp[Sum[A[-x^k]/k, {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = SeriesCoefficient[x Product[1/(1 - x^k)^((-1)^k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 0, 37}]

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} 1/(1 - x^n)^((-1)^n*a(n)).

Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^d*d*a(d) ) * a(n-k+1).

A307366 G.f. A(x) satisfies: A(x) = x*exp(A(-x) - A(-x^2)/2 + A(-x^3)/3 - A(-x^4)/4 + ...).

Original entry on oeis.org

0, 1, -1, 0, 0, 1, -2, -1, 3, 3, -8, -5, 17, 15, -47, -35, 118, 91, -311, -240, 839, 660, -2314, -1809, 6417, 5035, -18002, -14177, 51016, 40322, -145784, -115402, 419197, 332457, -1212617, -963586, 3526976, 2807301, -10307097, -8215194, 30246994, 24139050, -89101081

Offset: 0

Ilya Gutkovskiy, Apr 05 2019

sign

			G.f.: A(x) = x - x^2 + x^5 - 2*x^6 - x^7 + 3*x^8 + 3*x^9 - 8*x^10 - 5*x^11 + 17*x^12 + ...

Cf. A000081, A004111, A045648, A049075, A307365.

terms = 42; A[] = 0; Do[A[x] = x Exp[Sum[(-1)^(k + 1) A[-x^k]/k, {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = SeriesCoefficient[x Product[(1 + x^k)^((-1)^k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 0, 42}]

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + x^n)^((-1)^n*a(n)).

Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+d+1)*d*a(d) ) * a(n-k+1).

A036368 Number of chiral orthoplex n-ominoes in n-2 space.

Original entry on oeis.org

0, 0, 4, 14, 37, 110, 324, 888, 2368, 6336, 16874, 44414, 116181, 303362, 790157, 2051880, 5317599, 13764133, 35586766, 91910082, 237183164, 611701614, 1576773162, 4062606255, 10463699696, 26942811809, 69358469092

Offset: 4

Robert A. Russell

Orthoplex polyominoes are multidimensional polyominoes that do not extend more than two units along any axis.

			a(6)=4 because there are 4 pairs of chiral hexominoes in 2^4 space.

Cf. A045648, A036367.

sc[ n_, k_ ] := sc[ n, k ]=c[ n+1-k, 1 ]+If[ n<2k, 0, sc[ n-k, k ](-1)^k ]; c[ 1, 1 ] := 1;
c[ n_, 1 ] := c[ n, 1 ]=Sum[ c[ i, 1 ]sc[ n-1, i ]i, {i, 1, n-1} ]/(n-1);
c[ n_, k_ ] := c[ n, k ]=Sum[ c[ i, 1 ]c[ n-i, k-1 ], {i, 1, n-1} ];
Table[ c[ i, 4 ]/8+Sum[ c[ i, j ], {j, 5, i} ]/2-If[ OddQ[ i ], 0,
c[ i/2, 2 ](-1)^(i/2)/8+If[ OddQ[ i/2 ], 0, c[ i/4, 1 ](-1)^(i/4)/4 ]
+Sum[ c[ i/2, j ](-1)^(i/2), {j, 3, i/2} ]/2 ]+Sum[ c[ j, 1 ]c[ i-2j, 2 ](-1)^j/4
-Sum[ If[ OddQ[ k ], c[ j, (k-1)/2 ]c[ i-2j, 1 ](-1)^j/2, 0 ], {k, 5, i} ],
{j, 1, (i-1)/2} ], {i, 4, 30} ]

G.f.: (C^2(x) + C(-x^2))^2/8 - C^2(-x^2)/4 - C(-x^4)/4 + C^5(x)/(2-2C(x)) - (C(x)+C(-x^2))*C^2(-x^2)/(2-2C(-x^2)) where C(x) is the generating function for chiral n-ominoes in n-1 space, one cell labeled in A045648.

A345235 G.f. A(x) satisfies: A(x) = x + x^2 * exp(A(x) + A(-x^2)/2 + A(x^3)/3 + A(-x^4)/4 + ...).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 8, 14, 25, 44, 78, 142, 261, 479, 886, 1655, 3105, 5843, 11043, 20965, 39938, 76285, 146123, 280691, 540475, 1042885, 2016481, 3906647, 7582034, 14739395, 28697969, 55958110, 109262713, 213619535, 418158580, 819491034, 1607764395, 3157551026, 6207346544

Offset: 1

Ilya Gutkovskiy, Jun 11 2021

nonn

Cf. A007560, A007562, A045648, A345234.

nmax = 40; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[A[(-1)^(k + 1) x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; a[n_] := a[n] = (1/(n - 2)) Sum[(-1)^k Sum[(-1)^d d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 40}]

G.f.: x + x^2 / Product_{n>=1} (1 - (-x)^n)^((-1)^n*a(n)).
a(n+2) = (1/n) * Sum_{k=1..n} (-1)^k * ( Sum_{d|k} (-1)^d * d * a(d) ) * a(n-k+2).
a(n) ~ c * d^n / n^(3/2), where d = 2.04187801797233390910633071122033289228232310618876458... and c = 0.624667034123125135463988884805660643637934291759335... - Vaclav Kotesovec, Jun 19 2021

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A049075 Eigensequence of a power series transformation.

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A355049 Number of chiral pairs of orthoplex n-ominoes with cell centers determining n-3 space.

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A045649 Number of chiral n-ominoes in n-1 space.

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A355054 Number of chiral pairs of multidimensional n-ominoes with cell centers determining n-3 space.

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A036365 Number of chiral n-ominoes in n-2 space.

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A306768 G.f. A(x) satisfies: A(x) = x*exp(-A(-x) + A(-x^2)/2 - A(-x^3)/3 + A(-x^4)/4 - A(-x^5)/5 + ...).

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A307365 G.f. A(x) satisfies: A(x) = x*exp(A(-x) + A(-x^2)/2 + A(-x^3)/3 + A(-x^4)/4 + ...).

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A307366 G.f. A(x) satisfies: A(x) = x*exp(A(-x) - A(-x^2)/2 + A(-x^3)/3 - A(-x^4)/4 + ...).

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