A007808 -id:A007808 - cp's OEIS frontend

A141828 a(n) = (n^4*a(n-1)-1)/(n-1) for n >= 2, with a(0) = 1, a(1) = 5.

1, 5, 79, 3199, 272981, 42653281, 11055730435, 4424134795739, 2588750874763849, 2123099311165701661, 2358999234628557401111, 3453810779419670890966615, 6510747302004208690462157149, 15496121141045183700690805861049

Offset: 0

Peter Bala, Jul 09 2008

For related recurrences of the form a(n) = (n^k*a(n-1)-1)/(n-1) see A001339, A007808 (both k = 2) and A141827 (k = 3). a(n) is a difference divisibility sequence, that is, the difference a(n) - a(m) is divisible by n - m for all n and m (provided n is not equal to m). See A000522 for further properties of difference divisibility sequences.

Harvey P. Dale, Table of n, a(n) for n = 0..181

Cf. A001339, A007808, A141827.

a := n -> n!^3*add((n-k+1)*(k^2+k+1)/k!^3, k = 0..n): seq(a(n), n = 0..16);

nxt[{n_,a_}]:={n+1,((n+1)^4*a-1)/n}; Join[{1},NestList[nxt,{1,5},15][[All,2]]] (* Harvey P. Dale, Mar 12 2017 *)

Sum_{n>=0} a(n)*x^n/n!^3 = (1/(1-x)^2)*Sum_{n>=0} (n^2+n+1)*x^n/n!^3.
a(n) = n!^3*Sum_{k=0..n} (n-k+1)*(k^2+k+1)/k!^3.
a(n) = n*n!^3*(5 - Sum_{k=2..n} 1/(k!^3*k*(k-1))) for n > 0. [corrected by Jason Yuen, Jan 31 2025]
Congruence property: a(n) == (1+n+n^2+n^3) (mod n^4).
The recurrence a(n) = (n^3+n^2+n+2)*a(n-1) - (n-1)^3*a(n-2), n >= 2, shows that a(n) is always a positive integer. The sequence b(n) := n*n!^3 also satisfies the same recurrence with b(0) = 0, b(1) = 1. Hence we obtain the finite continued fraction expansion a(n)/(n*n!^3) = 5 - 1^3/(16 - 2^3/(41 - 3^3/(86 -...-(n-1)^3/(n^3+n^2+n+2)))), for n >= 1. a(n)*b(n+1) - b(n)*a(n+1) = n!^3.
Limit_{n->oo} a(n)/(n*n!^3) = Sum_{n>=0} (n^2+n+1)/n!^3 = 4.9367223378... .
Limit_{n->oo} a(n)/(n*n!^3) = 1 + Sum_{n>=0} 1/(Product_{k=0..n} A008620(k)).

A121298 Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n and height k (1<=k<=n; here by the height of a polyomino one means the number of lines of slope -1 that pass through the centers of the polyomino cells).

Original entry on oeis.org

1, 0, 2, 0, 1, 4, 0, 0, 5, 8, 0, 0, 3, 15, 16, 0, 0, 1, 17, 39, 32, 0, 0, 0, 15, 59, 95, 64, 0, 0, 0, 9, 75, 175, 223, 128, 0, 0, 0, 4, 78, 269, 479, 511, 256, 0, 0, 0, 1, 67, 358, 845, 1247, 1151, 512, 0, 0, 0, 0, 48, 419, 1300, 2461, 3135, 2559, 1024, 0, 0, 0, 0, 29, 432, 1801, 4224, 6813, 7679, 5631, 2048

Offset: 1

Emeric Deutsch, Aug 04 2006

Row sums are the odd-subscripted Fibonacci numbers (A001519). Sum of terms in column k = A007808(k). Sum(k*T(n,k),k=0..n)=A121299(n).

			T(2,2)=2 because we have the vertical and the horizontal dominoes.
Triangle starts:
1;
0,2;
0,1,4;
0,0,5,8;
0,0,3,15,16;
0,0,1,17,39,32;

E. Barcucci, A. Del Lungo, R. Pinzani and R. Sprugnoli, La hauteur des polyominos dirigés verticalement convexes, Actes du 31e Séminaire Lotharingien de Combinatoire, Publi. IRMA, Université Strasbourg I (1993).
E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.

Cf. A001519, A007808, A121299.

T:=proc(n,k) if n<=0 or k<=0 then 0 elif n=1 and k=1 then 1 else T(n-1,k-1)+add(T(n-k,j),j=1..k-1)+add(T(n-j,k-1),j=1..k-1) fi end: for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

T[n_, k_] := T[n, k] = Which[n <= 0 || k <= 0, 0, n == 1 && k == 1, 1, True, T[n-1, k-1] + Sum[T[n-k, j], {j, 1, k-1}] + Sum[T[n-j, k-1], {j, 1, k-1}]];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 25 2024, after Maple program *)

T(n,k) = T(n-1,k-1)+Sum(T(n-k,j), j=1..k-1)+Sum(T(n-j,k-1), j=1..k-1).

A136129 Triangle read by rows: T(n,k) is the number of directed, vertically convex polyominoes of height n and area k (n<= k <=n(n+1)/2).

Original entry on oeis.org

1, 0, 2, 1, 0, 0, 4, 5, 3, 1, 0, 0, 0, 8, 15, 17, 15, 9, 4, 1, 0, 0, 0, 0, 16, 39, 59, 75, 78, 67, 48, 29, 14, 5, 1, 0, 0, 0, 0, 0, 32, 95, 175, 269, 358, 419, 432, 400, 334, 250, 166, 97, 49, 20, 6, 1, 0, 0, 0, 0, 0, 0, 64, 223, 479, 845, 1300, 1801, 2269, 2622, 2805, 2794, 2593

Offset: 1

Emeric Deutsch, Jan 21 2008

Row n contains n(n+1)/2 terms. Row sums yield A007808. Column sums yield the odd-indexed Fibonacci numbers (A001519).

			Triangle starts:
1;
0,2,1;
0,0,4,5,3,1;
0,0,0,8,15,17,15,9,4,1;
0,0,0,0,16,39,59,75,78,67,48,29,14,5,1;

E. Barcucci, A. Del Lungo, R. Pinzani and R. Sprugnoli, La hauteur des polyominos dirigés verticalement convexes, Séminaire Lotharingien de Combinatoire, B31d (1993), 11 pp.

Cf. A007808, A001519.

A:=t*z*(1-t)/(1-t-2*t*z+t^2*z): B:=t^2*z*(z-1)/((1-t-2*t*z+t^2*z)*(1-t*z)): Aser:=simplify(series(A,z=0,12)): Bser:=simplify(series(B,z=0,12)): for n to 12 do A[n]:=coeff(Aser,z,n): B[n]:=coeff(Bser,z,n) end do: P[1]:=A[1]: for n from 2 to 7 do P[n]:=sort(expand(simplify(A[n]+add(B[n-j]*P[j]*t^j,j=1..n-1)))) end do: for n to 7 do seq(coeff(P[n],t,j),j=1..(1/2)*n*(n+1)) end do;

G.f. G(t,z) satisfies G(t,z)=zt(1-t)/(1-t-2zt+zt^2) +z(z-1)t^2*G(t,tz)/[(1-t-2zt+zt^2)(1-zt)]

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A141828 a(n) = (n^4*a(n-1)-1)/(n-1) for n >= 2, with a(0) = 1, a(1) = 5.

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A121298 Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n and height k (1<=k<=n; here by the height of a polyomino one means the number of lines of slope -1 that pass through the centers of the polyomino cells).

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A136129 Triangle read by rows: T(n,k) is the number of directed, vertically convex polyominoes of height n and area k (n<= k <=n(n+1)/2).

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