A003471 -id:A003471 - cp's OEIS frontend

A138464 Triangle read by rows: T(n, k) is the number of forests on n labeled nodes with k edges. T(n, k) for n >= 1 and 0 <= k <= n-1.

1, 1, 1, 1, 3, 3, 1, 6, 15, 16, 1, 10, 45, 110, 125, 1, 15, 105, 435, 1080, 1296, 1, 21, 210, 1295, 5250, 13377, 16807, 1, 28, 378, 3220, 18865, 76608, 200704, 262144, 1, 36, 630, 7056, 55755, 320544, 1316574, 3542940, 4782969, 1, 45, 990, 14070, 143325, 1092105, 6258000, 26100000, 72000000, 100000000

Offset: 1

N. J. A. Sloane, May 09 2008

The rows of the triangle give the coefficients of the Ehrhart polynomials of integral Coxeter permutahedra of type A. These polynomials count lattice points in a dilated lattice polytope. For a definition see Ardila et al. (p. 1158), the generating functions of these polynomials for the classical root systems are given in theorem 5.2 (p. 1163). - Peter Luschny, May 01 2021

			Triangle begins:
[1]  1;
[2]  1,  1;
[3]  1,  3,   3;
[4]  1,  6,  15,   16;
[5]  1, 10,  45,  110,  125;
[6]  1, 15, 105,  435, 1080,  1296;
[7]  1, 21, 210, 1295, 5250, 13377, 16807;

Alois P. Heinz, Rows n = 1..141, flattened
Federico Ardila, Matthias Beck, and Jodi McWhirter, The arithmetic of Coxeter permutahedra, Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. 44(173):1152-1166, 2020.
John Riordan and N. J. A. Sloane, Correspondence, 1974

Row sums give A001858. Rightmost diagonal gives A000272. Cf. A136605.
Rows reflected give A105599. - Alois P. Heinz, Oct 28 2011
Cf. A088956.
Lower diagonals give: A083483, A239910, A240681, A240682, A240683, A240684, A240685, A240686, A240687. - Alois P. Heinz, Apr 11 2014
T(2n,n) gives A302112.
For Ehrhart polynomials of integral Coxeter permutahedra of classical type cf. this sequence (type A), A343805 (type B), A343806 (type C), A343807 (type D).

T:= proc(n) option remember; if n=0 then 0 else T(n-1) +n^(n-1) *x^n/n! fi end: TT:= proc(n) option remember; expand(T(n) -T(n)^2/2) end: f:= proc(k) option remember; if k=0 then 1 else unapply(f(k-1)(x) +x^k/k!, x) fi end: A:= proc(n,k) option remember; series(f(k)(TT(n)), x,n+1) end: aa:= (n,k)-> coeff(A(n,k), x,n) *n!: a:= (n,k)-> aa(n,n-k) -aa(n,n-k-1): seq(seq(a(n,k), k=0..n-1), n=1..10);  # Alois P. Heinz, Sep 02 2008
alias(W = LambertW): EhrA := exp(-W(-t*x)/t - W(-t*x)^2/(2*t)):
ser := series(EhrA, x, 12): cx := n -> n!*coeff(ser, x, n):
T := n -> seq(coeff(cx(n), t, k), k=0..n-1):
seq(T(n), n = 1..10); # Peter Luschny, Apr 30 2021

t[0, 0] = 1; t[n_ /; n >= 1, k_] /; (0 <= k <= n-1) := t[n, k] = Sum[(i+1)^(i-1)*Binomial[n-1, i]*t[n-i-1, k-i], {i, 0, k}]; t[, ] = 0; Table[t[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jan 14 2014, after Peter Bala *)
gf := E^(-(ProductLog[-(t x)] (2 + ProductLog[-(t x)]))/(2 t));
ser := Series[gf, {x, 0, 12}]; cx[n_] := n! Coefficient[ser, x, n];
Table[CoefficientList[cx[n], t], {n, 1, 10}] // Flatten  (* Peter Luschny, May 01 2021 *)

From Peter Bala, Aug 14 2012: (Start)
T(n+1,k) = Sum_{i=0..k} (i+1)^(i-1)*binomial(n,i)*T(n-i,k-i) with T(0,0)=1.
Recurrence equation for row polynomials R(n,t): R(n,t) = Sum_{k=0..n-1} (k+1)^(k-1)*binomial(n-1,k)*t^k*R(n-k-1,t) with R(0,t) = R(1,t) = 1.
The production matrix for the row polynomials of the triangle is obtained from A088956 and starts:
    1    t
    1    1    t
    3    2    1    t
   16    9    3    1    t
  125   64   18    4    1   t
(End)
E.g.f.: exp( Sum_{n >= 1} n^(n-2)*t^(n-1)*x^n/n! ). - Peter Bala, Nov 08 2015
T(n, k) = [t^k] n! [x^n] exp(-W(-t*x)/t - W(-t*x)^2/(2*t)), where W denotes the Lambert function. - Peter Luschny, Apr 30 2021 [Typo corrected after note from Andrew Howroyd, Peter Luschny, Jun 20 2021]

More terms from Alois P. Heinz, Sep 02 2008

A000459 Number of multiset permutations of {1, 1, 2, 2, ..., n, n} with no fixed points.

Original entry on oeis.org

1, 0, 1, 10, 297, 13756, 925705, 85394646, 10351036465, 1596005408152, 305104214112561, 70830194649795010, 19629681235869138841, 6401745422388206166420, 2427004973632598297444857, 1058435896607583305978409166, 526149167104704966948064477665

Offset: 0

N. J. A. Sloane

Original definition: Number of permutations with no hits on 2 main diagonals. (Identical to definition of A000316.) - M. F. Hasler, Sep 27 2015
Card-matching numbers (Dinner-Diner matching numbers): A deck has n kinds of cards, 2 of each kind. The deck is shuffled and dealt in to n hands with 2 cards each. A match occurs for every card in the j-th hand of kind j. A(n) is the number of ways of achieving no matches. The probability of no matches is A(n)/((2n)!/2!^n).
Also, Penrice's Christmas gift numbers (see Penrice 1991).
a(n) is the maximal number of totally mixed Nash equilibria in games of n players, each with 3 pure options. - Raimundas Vidunas, Jan 22 2014

			There are 297 ways of achieving zero matches when there are 2 cards of each kind and 4 kinds of card so a(4)=297.
From _Peter Bala_, Jul 08 2014: (Start)
a(3) = 10: the 10 permutations of the multiset {1,1,2,2,3,3} that have no fixed points are
{2,2,3,3,1,1}, {3,3,1,1,2,2}
{2,3,1,3,1,2}, {2,3,1,3,2,1}
{2,3,3,1,1,2}, {2,3,3,1,2,1}
{3,2,1,3,1,2}, {3,2,1,3,2,1}
{3,2,3,1,1,2}, {3,2,3,1,2,1}
(End)

F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174-178.
R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
Per Alexandersson and Frether Getachew, An involution on derangements, arXiv:2105.08455 [math.CO], 2021.
Shalosh B. Ekhad, Christoph Koutschan, and Doron Zeilberger, There are EXACTLY 1493804444499093354916284290188948031229880469556 Ways to Derange a Standard Deck of Cards (ignoring suits) [and many other such useful facts], arXiv:2101.10147 [math.CO], 2021.
F. F. Knudsen and I. Skau, On the Asymptotic Solution of a Card-Matching Problem, Mathematics Magazine 69 (1996), 190-197.
P. A. MacMahon, Combinatory Analysis Cambridge: The University Press 1915-1916
B. H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), 107-118.
B. H. Margolius, Dinner-Diner Matching Probabilities
R. D. McKelvey and A. McLennan, The maximal number of regular totally mixed Nash equilibria, J. Economic Theory, 72:411--425, 1997.
L. I. Nicolaescu, Derangements and asymptotics of the Laplace transforms of large powers of a polynomial, New York J. Math. 10 (2004) 117-131.
S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98 (1991), 617-620.
John Riordan and N. J. A. Sloane, Correspondence, 1974
R. Vidunas, Counting derangements and Nash equilibria, arXiv preprint arXiv:1401.5400 [math.CO], 2014-2016.
Raimundas Vidunas, Counting derangements and Nash equilibria Ann. Comb. 21, No. 1, 131-152 (2017).
Wikipedia, Permutations of multisets
Index entries for sequences related to card matching

Cf. A000166, A008290, A059056-A059071, A033581, A059073.

Row n=2 of A372307.

I:=[0,1]; [n le 2 select I[n] else n*(2*n-1)*Self(n-1)+2*n*(n-1)*Self(n-2)-(2*n-1): n in [1..30]]; // Vincenzo Librandi, Sep 28 2015

p := (x,k)->k!^2*sum(x^j/((k-j)!^2*j!),j=0..k); R := (x,n,k)->p(x,k)^n; f := (t,n,k)->sum(coeff(R(x,n,k),x,j)*(t-1)^j*(n*k-j)!,j=0..n*k); seq(f(0,n,2)/2!^n,n=0..18);

RecurrenceTable[{(2*n+3)*a[n+3]==(2*n+5)^2*(n+2)*a[n+2]+(2*n+3)*(n+2)*a[n+1]-2*(2*n+5)*(n+1)*(n+2)*a[n],a[1]==0,a[2]==1,a[3]==10},a,{n,1,25}] (* Vaclav Kotesovec, Aug 31 2012 *)
a[n_] := a[n] = n*(2*n-1)*a[n-1] + 2*n*(n-1)*a[n-2] - (2*n-1); a[0] = 1; a[1] = 0; a[2] = 1; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Mar 04 2013 *)
a[n_] := Sum[(2*(n-m))! / 2^(n-m) Binomial[n, m] Hypergeometric1F1[m-n, 2*(m - n), -4], {m, 0, n}]; Table[a[n], {n, 0, 16}] (* Peter Luschny, Nov 15 2023 *)

a(n) = (2^n*round(2^(n/2+3/4)*Pi^(-1/2)*exp(-2)*n!*besselk(1/2+n,2^(1/2))))/2^n;
vector(15, n, a(n))\\ Altug Alkan, Sep 28 2015

{ A000459(n) = sum(m=0,n, sum(k=0,n-m, (-1)^k * binomial(n,k) * binomial(n-k,m) * 2^(2*k+m-n) * (2*n-2*m-k)! )); } \\ Max Alekseyev, Oct 06 2016

a(n) = A000316(n)/2^n.
a(n) = Sum_{k=0..n} Sum_{m=0..n-k} (-1)^k * n!/(k!*m!*(n-k-m)!) * 2^(2*k+m-n) * (2*n-2*m-k)!. - Max Alekseyev, Oct 06 2016
G.f.: Sum_{j=0..n*k} coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)! where n is the number of kinds of cards, k is the number of cards of each kind (2 in this case) and coeff(R(x, n, k), x, j) is the coefficient of x^j of the rook polynomial R(x, n, k) = (k!^2*sum(x^j/((k-j)!^2*j!))^n (see Riordan or Stanley).
D-finite with recurrence a(n) = n*(2*n-1)*a(n-1)+2*n*(n-1)*a(n-2)-(2*n-1), a(1) = 0, a(2) = 1.
a(n) = round(2^(n/2 + 3/4)*Pi^(-1/2)*exp(-2)*n!*BesselK(1/2+n,2^(1/2))). - Mark van Hoeij, Oct 30 2011
(2*n+3)*a(n+3)=(2*n+5)^2*(n+2)*a(n+2)+(2*n+3)*(n+2)*a(n+1)-2*(2*n+5)*(n+1)*(n+2)*a(n). - Vaclav Kotesovec, Aug 31 2012
Asymptotic: a(n) ~ n^(2*n)*2^(n+1)*sqrt(Pi*n)/exp(2*n+2), Vaclav Kotesovec, Aug 31 2012
a(n) = (1/2^n)*A000316(n) = int_{0..inf} exp(-x)*(1/2*x^2 - 2*x + 1)^n dx. Asymptotic: a(n) ~ ((2*n)!/2^n)*exp(-2)*( 1 - 1/(2*n) - 23/(96*n^2) + O(1/n^3) ). See Nicolaescu. - Peter Bala, Jul 07 2014
Let S = x_1 + ... + x_n. a(n) equals the coefficient of (x_1*...*x_n)^2 in the expansion of product {i = 1..n} (S - x_i)^2 (MacMahon, Chapter III). - Peter Bala, Jul 08 2014
Conjecture: a(n+k) - a(n) is divisible by k. - Mark van Hoeij, Nov 15 2023

More terms and edited by Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 22 2000
Edited by M. F. Hasler, Sep 27 2015
a(0)=1 prepended by Max Alekseyev, Oct 06 2016

A003470 a(n) = n*a(n-1) - a(n-2) + 1 + (-1)^n.

Original entry on oeis.org

1, 1, 3, 8, 31, 147, 853, 5824, 45741, 405845, 4012711, 43733976, 520795003, 6726601063, 93651619881, 1398047697152, 22275111534553, 377278848390249, 6768744159489931, 128228860181918440, 2557808459478878871, 53585748788874537851, 1176328664895760953853

Offset: 0

N. J. A. Sloane and John Riordan

Row sums of A086764. - Philippe Deléham, Apr 27 2004

a(n+2m) == a(n) (mod m) for all n and m. - Robert Israel, Dec 06 2016

			G.f. = 1 + x + 3*x^2 + 8*x^3 + 31*x^4 + 147*x^5 + 853*x^6 + 5824*x^7 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
John Riordan and N. J. A. Sloane, Correspondence, 1974

Cf. A008279, A072374, A086764, A143409.

f:= gfun:-rectoproc({a(n) -(n-1)*a(n-1)-(n-2)*a(n-2)+a(n-3)-2=0,a(0)=1,a(1)=1,a(2)=3},a(n),remember):
map(f, [$0..30]); # Robert Israel, Dec 06 2016

t = {1, 1}; Do[AppendTo[t, n*t[[-1]] - t[[-2]] + 1 + (-1)^n], {n, 2, 20}] (* T. D. Noe, Oct 07 2013 *)
T[n_, k_] := HypergeometricPFQ[{k+1, k-n},{},-1];
Table[Sum[(-1)^k T[n,k], {k,0,n}], {n,0,22}] (* Peter Luschny, Oct 05 2017 *)

Diagonal sums of reverse of permutation triangle A008279. a(n) = Sum_{k=0..floor(n/2)} (n-k)!/k!. - Paul Barry, May 12 2004
a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*(n-2k)!. - Paul Barry Dec 15 2010
G.f.: 1/(1-x^2-x/(1-x/(1-x^2-2x/(1-2x/(1-x^2-3x/(1-3x/(1-x^2-4x/(1-4x/(1-.... (continued fraction);
G.f.: 1/(1-x-x^2-x^2/(1-3x-x^2-4x^2/(1-5x-x^2-9x^2/(1-7x-x^2-16x^2/(1-... (continued fraction). - Paul Barry, Dec 15 2010
G.f.: hypergeom([1,1],[],x/(1-x^2))/(1-x^2). - Mark van Hoeij, Nov 08 2011
G.f.: 1/Q(0), where Q(k)= 1 - x^2 - x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 20 2013
From Robert Israel, Dec 06 2016: (Start)
a(2m) = hypergeom([1,-m,m+1],[],-1).
a(2m+1) = hypergeom([1,-m,m+2],[],-1)*(m+1).
a(2m-1) + a(2m+1) = (2m+1) a(2m). (End)
0 = a(n)*(-a(n+2) - a(n+3)) + a(n+1)*(-2 + a(n+1) - 2*a(n+3) + a(n+4)) + a(n+2)*(-2*a(n+3) + a(n+4)) + a(n+3)*(+2 - a(n+3)) if n >= 0. - Michael Somos, Dec 06 2016
0 = a(n)*(-a(n+2) + a(n+4)) + a(n+1)*(+a(n+1) - a(n+2) - a(n+3) + 3*a(n+4) - a(n+5)) + a(n+2)*(-a(n+3) + a(n+4)) + a(n+3)*(-a(n+4) + a(n+5)) + a(n+4)*(-a(n+4)) if n >= 0. - Michael Somos, Dec 06 2016
a(n) = Sum_{k=0..n} (-1)^k*hypergeom([k+1, k-n], [], -1). - Peter Luschny, Oct 05 2017
D-finite with recurrence: a(n) -n*a(n-1) +(n-2)*a(n-3) -a(n-4)=0. - R. J. Mathar, Apr 29 2020
a(n) ~ n! * (1 + 1/n + 1/(2*n^2) + 2/(3*n^3) + 25/(24*n^4) + 77/(40*n^5) + 2971/(720*n^6) + 6287/(630*n^7) + 1074809/(40320*n^8) + 28160749/(362880*n^9) + ...). - Vaclav Kotesovec, Nov 25 2022

More terms from Gabriel Cunningham (gcasey(AT)mit.edu), Oct 25 2004

A000316 Two decks each have n kinds of cards, 2 of each kind. The first deck is laid out in order. The second deck is shuffled and laid out next to the first. A match occurs if a card from the second deck is next to a card of the same kind from the first deck. a(n) is the number of ways of achieving no matches.

Original entry on oeis.org

1, 0, 4, 80, 4752, 440192, 59245120, 10930514688, 2649865335040, 817154768973824, 312426715251262464, 145060238642780180480, 80403174342119992692736, 52443098500204184915312640, 39764049487996490505336537088

Offset: 0

N. J. A. Sloane

Each deck contains 2n cards.
The probability of no matches is a(n)/(2n)!.
n couples meet for a party and they exchange gifts. Each of the 2n writes their name on a piece of paper and puts it into a hat. They take turns drawing names and give their gift to the person whose name they drew. If anyone draws either their own name or the name of their partner, everyone puts the name they have drawn back into the hat and everyone draws anew. a(n) is the number of different permissible drawings. - Dan Dima, Dec 17 2006
(2n)! / a(n) is the expected number of deck shuffles until no matches occur. a(n) / (2n)! is the probability for a permissible drawing to be achieved. (2n)! / a(n) is the expected number of drawings before a permissible drawing is achieved. As n goes to infinity (2n)! / a(n) will strictly decrease very slowly to e^2 ~ 7.38906 (starting from n > 2) - Dan Dima, Dec 17 2006
a(n) equals the permanent of the (2n)X(2n) matrix with 0's along the main diagonal and the antidiagonal, and 1's everywhere else. - John M. Campbell, Jul 11 2011
Also, number of permutations p of (1,...,2n) such that round(p(k)/2) != round(k/2) for all k=1,...,2n (where half-integers are rounded up). - M. F. Hasler, Sep 30 2015

			There are 80 ways of achieving zero matches when there are 2 cards of each kind and 3 kinds of card so a(3)=80.
Among the 24 (multiset) permutations of {1,1',2,2'}, only {2,2',1,1'}, {2',2,1,1'}, {2,2',1',1} and {2',2,1',1} have no fixed points, thus a(2)=4.

F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 187.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
F. F. Knudsen and I. Skau, On the Asymptotic Solution of a Card-Matching Problem, Mathematics Magazine 69 (1996), 190-197.
B. H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), 107-118.
Barbara H. Margolius, Dinner-Diner Matching Probabilities
L. I. Nicolaescu, Derangements and asymptotics of the Laplace transforms of large powers of a polynomial, New York J. Math. 10 (2004) 117-131.
John Riordan and N. J. A. Sloane, Correspondence, 1974
E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).
Index entries for sequences related to card matching

Cf. A000166, A000459, A337302.

p := (x,k)->k!^2*sum(x^j/((k-j)!^2*j!),j=0..k); R := (x,n,k)->p(x,k)^n; f := (t,n,k)->sum(coeff(R(x,n,k),x,j)*(t-1)^j*(n*k-j)!,j=0..n*k); seq(f(0,n,2), n=0..18);

(* b = A000459 *)
b[n_] := b[n] = Switch[n, 0, 1, 1, 0, 2, 1, _, n(2n-1) b[n-1] + 2n(n-1) b[n-2] - (2n-1)];
a[n_] := b[n] * 2^n;
Array[a, 14] (* Jean-François Alcover, Oct 30 2019 *)

a(n)=if(n==0, 1, round(2^(n/2+3/4)/Pi^.5*exp(-2)*n!*besselk(1/2+n,2^.5))<M. F. Hasler, Sep 27 2015

\\ Illustration of the multiset-fixed-point interpretation
count(n,c=ceil(vector(n,i,i)/2))=sum(k=1,n!,!setsearch(Set(ceil(Vec(numtoperm(n,k))/2)-c),0))
a(n) = count(2*n) \\ M. F. Hasler, Sep 30 2015

a(n) = A000459(n)*2^n.
G.f.: Sum_{j=0..n*k} coeff(R(x, n, k), x, j)*(-1)^j*(n*k-j)! where n is the number of kinds of cards, k is the number of cards of each kind (2 in this case) and R(x, n, k) is the rook polynomial given by R(x, n, k)=(k!^2*Sum_{j=0..k} x^j/((k-j)!^2*j!))^n (see Riordan). coeff(R(x, n, k), x, j) indicates the coefficient for x^j of the rook polynomial.
From Dan Dima, Dec 17 2006: (Start)
a(n) = n! * Sum_{a,b >= 0, a+b <= n} (-1)^b * 2^(a+2*b) * (2*n-2*a-b)! / (a! * b! * (n-a-b)!).
a(n) = n * a(n-1) + n! * 4^n * Sum_{a=0..n} (-1)^a / (a! * 2^a). (End)
a(n) = 2^n * round(2^(n/2 + 3/4)*Pi^(-1/2)*exp(-2)*n!*BesselK(1/2+n,2^(1/2))) for n > 0. - Mark van Hoeij, Oct 30 2011
Recurrence: (2*n-3)*a(n) = 2*(n-1)*(2*n-1)^2*a(n-1) + 4*(n-1)*(2*n-3)*a(n-2) - 16*(n-2)*(n-1)*(2*n-1)*a(n-3). - Vaclav Kotesovec, Aug 07 2013
From Peter Bala, Jul 07 2014: (Start)
a(n) = Integral_{x>=0} exp(-x)*(x^2 - 4*x + 2)^n dx. Cf. A000166(n) = Integral_{x>=0} exp(-x)*(x - 1)^n dx.
Asymptotic: a(n) ~ (2*n)!*exp(-2)*( 1 - 1/(2*n) - 23/(96*n^2) + O(1/n^3) ). See Nicolaescu. (End)

Formulae, more terms etc. from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 22 2000
Edited by M. F. Hasler, Sep 27 2015 and N. J. A. Sloane, Oct 02 2015
a(0)=1 prepended by Andrew Howroyd, Oct 09 2020

A003482 a(n) = 7*a(n-1) - a(n-2) + 4, with a(0) = 0, a(1) = 5.

Original entry on oeis.org

0, 5, 39, 272, 1869, 12815, 87840, 602069, 4126647, 28284464, 193864605, 1328767775, 9107509824, 62423800997, 427859097159, 2932589879120, 20100270056685, 137769300517679, 944284833567072, 6472224534451829, 44361286907595735, 304056783818718320

Offset: 0

N. J. A. Sloane

The values (a(n),x(n)), n >= 2, x(n)=Fibonacci(2*n+2)*Fibonacci(2*n+3)=A081018(n+1), are the integer solutions (a,x) of the equation binomial(x+1,a+1) + binomial(x+2,a+1) = binomial(x+3,a+1). - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

The values (a(n),x(n)), n >= 2 are also the integer solutions (a, x) of the equation x(a+1) = (x-a)(x-a-1) or, equivalently, binomial(x, a) = binomial(x-1, a+1). - Tomohiro Yamada, May 30 2018

			G.f. = 5*x + 39*x^2 + 272*x^3 + 1869*x^4 + 12815*x^5 + 87840*x^6 + ... - _Michael Somos_, Jun 26 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Heiko Harborth, Fermat-like binomial equations, Applications of Fibonacci numbers, Proc. 2nd Int. Conf., San Jose, California, August 1986, 1-5 (1988).
Sébastien Labbé and Jana Lepšová, A Fibonacci's complement numeration system, arXiv:2205.02574 [cs.FL], 2022.
Sébastien Labbé and Jana Lepšová, A Fibonacci analogue of the two's complement numeration system, RAIRO-Theor. Inf. Appl. (2023) Vol. 57, No. 12. See p. 16.
D. A. Lind, The quadratic field Q(sqrt(5)) and a certain diophantine equation, Fibonacci Quart. 6(3) (1968), 86-93.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
John Riordan and N. J. A. Sloane, Correspondence, 1974
David Singmaster, Repeated binomial coefficients and Fibonacci numbers, Fibonacci Quart. 13 (1973), 295-298.
S. M. Tanny and M. Zuker, On a unimodal sequence of binomial coefficients, Discrete Math. 9 (1974), 79-89.
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000045, A001109, A001654, A081018, A206351.

A003482:=z*(-5+z)/(z-1)/(z**2-7*z+1); # conjectured by Simon Plouffe in his 1992 dissertation

LinearRecurrence[{8,-8,1},{0,5,39},30] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)
nxt[{a_,b_}]:={b,7b-a+4}; NestList[nxt,{0,5},30][[;;,1]] (* Harvey P. Dale, Jun 09 2025 *)

a(n)=fibonacci(2*n)*fibonacci(2*n+3) \\ Charles R Greathouse IV, May 29 2013

a(n) = Fibonacci(2*n) * Fibonacci(2*n+3).
a(n) = Fibonacci(2*n+2)^2 - Fibonacci(2*n+1)^2. - Gary Detlefs, Oct 12 2011
a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3). - Vladimir Joseph Stephan Orlovsky and Vincenzo Librandi, Jan 22 2012
a(n) = -4/5 + (sqrt(5)/5 + 2/5)*(7/2 + 3*sqrt(5)/2)^n - (sqrt(5)/5 - 2/5)*(7/2 - 3*sqrt(5)/2)^n. - Antonio Alberto Olivares, May 29 2013
a(n) = -A206351(-n) for all n in Z. - Michael Somos, Jun 26 2018
From Sébastien Labbé, May 06 2022: (Start)
a(n) = Sum_{k=2..2*n+1} Fibonacci(k)^2.
a(n) = A001654(2*n+1)-1. (End)

A059300 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 4.

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 12, 24, 4, 1, 20, 90, 80, 5, 1, 30, 240, 540, 240, 6, 1, 42, 525, 2240, 2835, 672, 7, 1, 56, 1008, 7000, 17920, 13608, 1792, 8, 1, 72, 1764, 18144, 78750, 129024, 61236, 4608, 9, 1, 90, 2880, 41160, 272160, 787500, 860160, 262440, 11520, 10

Offset: 0

N. J. A. Sloane, Jan 25 2001

			Triangle begins:
1;
1,  2;
1,  6,   3;
1, 12,  24,    4;
1, 20,  90,   80,    5;
1, 30, 240,  540,  240,   6;
1, 42, 525, 2240, 2835, 672, 7;
...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
John Riordan and N. J. A. Sloane, Correspondence, 1974

There are 4 versions: A059297-A059300. Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc. Row sums are A000248.

/* As triangle: */ [[Binomial(n+1,n-k+1)*(n-k+1)^k: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 22 2015

t[n_, k_] := Binomial[n + 1, k]*(n - k + 1)^k; Flatten@Table[t[n, k], {n, 0, 9}, {k, 0, n}] (* Arkadiusz Wesolowski, Mar 23 2013 *)

for(n=0, 25, for(k=0, n, print1(binomial(n+1,k)*(n-k+1)^k, ", "))) \\ G. C. Greubel, Jan 05 2017

T(n,k) = binomial(n+1,n-k+1)*(n-k+1)^k. - R. J. Mathar, Mar 14 2013

A001862 Number of forests of least height with n nodes.

Original entry on oeis.org

1, 1, 2, 7, 26, 111, 562, 3151, 19252, 128449, 925226, 7125009, 58399156, 507222535, 4647051970, 44747776651, 451520086208, 4761032807937, 52332895618066, 598351410294193, 7102331902602676, 87365859333294151, 1111941946738682522, 14621347433458883187

Offset: 0

N. J. A. Sloane

nonn

From Gus Wiseman, Feb 14 2024: (Start)
Also the number of minimal loop-graphs covering n vertices. This is the minimal case of A322661. For example, the a(0) = 1 through a(3) = 7 loop-graphs are (loops represented as singletons):
  {}  {1}  {12}   {1-23}
           {1-2}  {2-13}
                  {3-12}
                  {12-13}
                  {12-23}
                  {13-23}
                  {1-2-3}
(End)

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983. See (3.3.7): number of ways to cover the complete graph K_n with star graphs.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
John Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.
John Riordan, Letter to N. J. A. Sloane, Oct. 1970
John Riordan and N. J. A. Sloane, Correspondence, 1974

The connected case is A000272.
Without loops we have A053530, minimal case of A369191.
This is the minimal case of A322661.
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 counts simple graphs; also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
Cf. A000085, A000169, A003465, A062740, A066383, A133686, A368597.

Range[0, 20]! CoefficientList[Series[Exp[x Exp[x] - x^2/2], {x, 0, 20}], x] (* Geoffrey Critzer, Mar 13 2011 *)
fasmin[y_]:=Complement[y,Union@@Table[Union[s,#]& /@ Rest[Subsets[Complement[Union@@y,s]]],{s,y}]];
Table[Length@fasmin[Select[Subsets[Subsets[Range[n],{1,2}]], Union@@#==Range[n]&]],{n,0,4}] (* Gus Wiseman, Feb 14 2024 *)

E.g.f.: exp(x*(exp(x)-x/2)).

Binomial transform of A053530. - Gus Wiseman, Feb 14 2024

Formula and more terms from Vladeta Jovovic, Mar 27 2001

A245094 Total squares count in n-th generation of Pythagoras tree variation which is rhombitrihexagonal tiling.

Original entry on oeis.org

1, 2, 4, 8, 13, 20, 24, 27, 33, 36, 42, 45, 51, 54, 60, 63, 69, 72, 78, 81, 87, 90, 96, 99, 105, 108, 114, 117, 123, 126, 132, 135, 141, 144, 150, 153, 159, 162, 168, 171, 177, 180, 186, 189, 195, 198, 204, 207, 213, 216, 222, 225, 231, 234, 240, 243, 249, 252, 258, 261, 267

Offset: 0

Kival Ngaokrajang, Nov 12 2014

nonn

Refer to Pythagoras tree (fractal) in the link. In the article "Varying the angle", the construction rule is changed from the standard Pythagoras tree by changing the base angle from 90 degrees to 60 degrees. It is easily seen that the size of the unit squares remains constant and equal to sin(30 degrees)/(1/2) = 1. The first overlap occurs at the fifth generation (n=4). The general pattern produced is the rhombitrihexagonal tiling, an array of hexagons bordered by the constructing squares. a(n) gives total count of squares in n-th generation which excluding the overlap into (n-1)-th generation and count only 1 for the overlap among current one. See illustration.

Conjecture: In the limit n -> infinity this construction produces one of the eight planar semiregular tessellations (one of the 11 Archimedean tessellations, the other three being regular). This is the tessellation (3,4,6,4) because of the sequence of regular 3-, 4- and 6-gons around each vertex. See the Eric Weisstein link. - Wolfdieter Lang, Nov 23 2014

Kival Ngaokrajang, Illustration of initial terms
John Riordan and N. J. A. Sloane, Correspondence, 1974
Eric Weisstein's World of Mathematics, Semiregular Tessellation
Wikipedia, Pythagoras tree
Wikipedia, Rhombitrihexagonal tiling
Index entries for linear recurrences with constant coefficients, signature (1, 1, -1).

Cf. A226454, A227298.

a[n_] := a[n] = If[n <= 6, {1, 2, 4, 8, 13, 20, 24}[[n+1]], a[n-1] + 6 - 3 Mod[n, 2]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 24 2016, adapted from PARI *)

{a=24;print1("1, 2, 4, 8, 13, 20, ",a,", ");
for (n=7,100,if (Mod(n,2)==1,d1=3,d1=6);a=a+d1;print1(a,", "))}

Conjectures from Colin Barker, Nov 12 2014: (Start)
a(n) = 3*((-1)^n + 6*n-5)/4 for n > 5.
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 8.
G.f.: (2*x^8 - 4*x^7 - x^6 + 3*x^5 + 3*x^4 + 3*x^3 + x^2 + x + 1) / ((x-1)^2*(x+1)).
(End)
It follows from the above conjecture that this sequence consists of interlaced polynomials for n > 5: a(2n) = 3*(3n-1) and a(2n+1) = 9*n. - Avi Friedlich, May 09 2015

A003481 a(n) = 7*a(n-1) - a(n-2) + 5.

Original entry on oeis.org

2, 20, 143, 986, 6764, 46367, 317810, 2178308, 14930351, 102334154, 701408732, 4807526975, 32951280098, 225851433716, 1548008755919, 10610209857722, 72723460248140, 498454011879263, 3416454622906706, 23416728348467684, 160500643816367087, 1100087778366101930

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
John Riordan and N. J. A. Sloane, Correspondence, 1974.
S. M. Tanny and M. Zuker, On a unimodal sequence of binomial coefficients, Discrete Math. 9 (1974), 79-89.
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A033888.

t = {2, 20}; Do[AppendTo[t, 7*t[[-1]] - t[[-2]] + 5], {n, 2, 30}] (* T. D. Noe, Oct 07 2013 *)
nxt[{a_,b_}]:={b,7b-a+5}; NestList[nxt,{2,20},30][[All,1]] (* Harvey P. Dale, Aug 11 2019 *)

G.f.: ( -2-4*x+x^2 ) / ( (x-1)*(x^2-7*x+1) ). - Simon Plouffe in his 1992 dissertation

a(n) = Fibonacci(4(n+1)) - 1 = A033888(n+1) - 1. - Ralf Stephan, Feb 24 2004, index corrected R. J. Mathar, Sep 18 2008

More terms from Ralf Stephan, Feb 24 2004

A002777 Restricted permutations.

Original entry on oeis.org

1, 0, 0, 0, 4, 16, 80, 672, 4896, 49920, 460032, 5598720, 62584320, 885381120, 11644323840, 187811205120, 2841958748160, 51481298534400, 881192033648640, 17714783352913920, 338434210452602880, 7477275543168614400

Offset: 0

N. J. A. Sloane

T. Muir, The Theory of Determinants in the Historical Order of Development. 4 vols., Macmillan, NY, 1906-1923, Vol. 3, p. 468.
Todd Simpson, Permutations with unique fixed and reflected points. Ars Combin. 39 (1995), 97-108.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923, Vol. 2.
T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923. [Annotated scans of selected pages]. See Vol. 3 page 468. There may have been some confusion here of this sequence with A003471.
T. Simpson, Letter to N. J. A. Sloane, Mar. 1992.
T. Simpson, Permutations with unique fixed and reflected points, Preprint. (Annotated scanned copy)

Cf. A003471.

a:= proc(n) option remember; `if`(n<5, [1, 0$3, 4][n+1],
      (n-1)*a(n-1)+2*`if`(n::even, (n-2)*a(n-3), (n-1)*a(n-2)))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 27 2020

nmax = 20; b = ConstantArray[0, nmax+1]; b[[1]] = 1; b[[2]] = 0; b[[3]] = 0; b[[4]] = 0; b[[5]] = 4; Do[b[[n+1]] = (n-1)*b[[n]] + If[EvenQ[n],2*(n-2)*b[[n-2]],2*(n-1)*b[[n-1]]], {n, 5, nmax}]; b (* Vaclav Kotesovec, Mar 07 2014 *)

a(n) = (n-1)*a(n-1) + 2*(n-d)*a(n-e), where (d, e) = (2, 3) if n even, (1, 2) if n odd.
Recurrence (for n>=7): (3*n^2 - 17*n + 23)*a(n) = (3*n^2 - 17*n + 21)*a(n-1) + (3*n^4 - 23*n^3 + 63*n^2 - 74*n + 34)*a(n-2) - 4*(n-3)*(n-2)*a(n-3) + 2*(n-4)*(n-3)*(3*n^2 - 11*n + 9)*a(n-4). - Vaclav Kotesovec, Mar 07 2014
a(n) ~ c * n!, where c = 5*sinh(sqrt(2))/2^(3/2) - 3*cosh(sqrt(2))/2 = 0.15347184510862040153106983922669125715345689997588202335369... - Vaclav Kotesovec, Mar 07 2014, updated Apr 20 2024

More terms from Larry Reeves (larryr(AT)acm.org), Sep 24 2001

cp's OEIS Frontend

A138464 Triangle read by rows: T(n, k) is the number of forests on n labeled nodes with k edges. T(n, k) for n >= 1 and 0 <= k <= n-1.

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A000459 Number of multiset permutations of {1, 1, 2, 2, ..., n, n} with no fixed points.

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A003470 a(n) = n*a(n-1) - a(n-2) + 1 + (-1)^n.

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A003482 a(n) = 7*a(n-1) - a(n-2) + 4, with a(0) = 0, a(1) = 5.

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A059300 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 4.

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