Richard Choulet - cp's OEIS frontend

A185106 Column 4 of A181783.

1, 7, 63, 709, 9709, 157971, 2993467, 64976353, 1593358809, 43632348319, 1321213523191, 43869502390077, 1585770335098693, 62013234471100459, 2609265444024424179, 117558236422872707161, 5647316731308685308337, 288166881285968665526583, 15566545814457889774570159, 887503412305357492886020789

Offset: 0

Richard Choulet, Dec 26 2012

A181783 is written as follows:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 2, 4, 7, 11, 16, ...
1, 1, 5, 21, 63, 151, 311, ...
1, 1, 16, 142, 709, 2521, ...
1, 1, 65, 1201, 9709, ...
A000522 and A053482 are respectively the columns number 2 and 3 of this array. Our sequence gives the column number 4 (the fifth).

G. C. Greubel, Table of n, a(n) for n = 0..375

Cf. A181783, A000522, A053482.

a(0,1):=1:for p from 2 to 15 do for n from 0 to 20 do a(n,0):=1 :a(n,p):=n!*sum('1/(n-m)!*sum('k^(p-2)*(-1)^(p-1-k)*k^m/((k-1)!*(p-1-k)!)','k'=1..(p-1))','m'=0..n):od:seq(a(n,p),n=0..20):od;

CoefficientList[Series[E^x/((1-x)*(1-2x)*(1-3x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 02 2013 *)

x='x+O('x^50); Vec(serlaplace(exp(x)/((1-x)*(1-2*x)*(1-3*x)))) \\ G. C. Greubel, Jun 22 2017

Recurrence relation: a(n)= 6*n*a(n-1) -11*n*(n-1)*a(n-2) +6*n*(n-1)*(n-2)*a(n-3) +1 (or following A053482 for a linear homogeneous recurrence) a(n)= (6n+1)*a(n-1) -(11n+6)*(n-1)*a(n-2) +(6n+11)*(n-1)*(n-2)*a(n-3) -6*(n-1)*(n-2)*(n-3)*a(n-4).
E.g.f: exp(z)/((1-z)*(1-2*z)*(1-3*z)), as explained in A181783.
With p=4, a(n)=a(n,p)=n!*sum('1/(n-m)!*sum('k^(p-2)*(-1)^(p-1-k)*k^m/((k-1)!*(p-1-k)!)','k'=1..(p-1))','m'=0..n)
a(n) ~ n! * exp(1/3)*3^(n+2)/2. - Vaclav Kotesovec, Oct 02 2013

A181783 Array described in comments to A053482, here read by increasing antidiagonals. See comments below.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 4, 1, 1, 1, 16, 21, 7, 1, 1, 1, 65, 142, 63, 11, 1, 1, 1, 326, 1201, 709, 151, 16, 1, 1, 1, 1957, 12336, 9709, 2521, 311, 22, 1, 1, 1, 13700, 149989, 157971, 50045, 7186, 575, 29, 1, 1, 1, 109601, 2113546, 2993467, 1158871, 193765, 17536, 981, 37, 1

Offset: 0

Richard Choulet, Dec 23 2012

We denote by a(n,k) the number in row number n >= 0 and column number k >= 0. The recurrence which defines the array is a(n,k) = n*(k-1)*a(n-1,k) + a(n,k-1). The initial values are given by a(n,0) = 1 = a(0,k) for all n >= 0 and k >= 0.

			Array read row after row:
  1, 1,    1,      1,       1,        1,         1, ...
  1, 1,    2,      4,       7,       11,        16, ...
  1, 1,    5,     21,      63,      151,       311, ...
  1, 1,   16,    142,     709,     2521,      7186, ...
  1, 1,   65,   1201,    9709,    50045,    193765, ...
  1, 1,  326,  12336,  157971,  1158871,   6002996, ...
  1, 1, 1957, 149989, 2993467, 30806371, 210896251, ...
  ...
A(4,3) = 1201.

Cf. A000522, A053482, A185106, A371898.

A181783 := proc(n,k)
    option remember;
    if n =0 or k = 0 then
        1;
    else
        n*(k-1)*procname(n-1,k)+procname(n,k-1) ;
    end if;
end proc:
seq(seq(A181783(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Mar 02 2016

T[n_, k_] := T[n, k] = If[n == 0 || k == 0, 1, n (k - 1) T[n - 1, k] + T[n, k - 1]];
Table[T[n - k, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 10 2023 *)

If we consider the e.g.f. Psi(k) of column number k we have: Psi(k)(z) = Psi(k-1)(z)/(1-(k-1)*z) with Psi(1)(z) = exp(z). Then Psi(k)(z) = exp(z)/Product_{j=0..k-1} (1 - j*z). We conclude that a(n,k) = n!*Sum_{m=0..n} Sum_{j=1..k-1} (-1)^(k-1-j)*j^(m+k-2)/((n-m)!*(j-1)!*(k-1-j)!). It seems after the recurrence (and its proof) in A053482 that:
A(n,k) = -Sum_{j=1..k-1} s1(k,k-j)*n*(n-1)*...*(n-k+1)*a(n-j,k) + 1 where s1(m,n) are the classical Stirling numbers of the first kind.
A(n,1) = 1 for every n.
A(1,k) = 1 + k*(k-1)/2 for every k.
A(n, k+1) = A371898(n+k, k) * n! / ((n+k)! * k!). - Werner Schulte, Apr 14 2024

Edited by N. J. A. Sloane, Dec 24 2012

A159035 a(0)=1=a(1), a(2)=2, a(3)=5; thereafter a(n+3)=4a(n+2)-4a(n+1)+2*a(n) for n>=1.

Original entry on oeis.org

1, 1, 2, 5, 14, 40, 114, 324, 920, 2612, 7416, 21056, 59784, 169744, 481952, 1368400, 3885280, 11031424, 31321376, 88930368, 252498816, 716916544, 2035531648, 5779458048, 16409538688, 46591385856, 132286304768, 375598753024

Offset: 0

Richard Choulet, Apr 03 2009

A117189 prefixed by an initial 1; essentially a duplicate. - N. J. A. Sloane and R. J. Mathar, Apr 07 2009

G.f.: f(z)=((1-3*z+2*z^2-z^3)/(1-4*z+4*z^2-2*z^3))

A174837 Sequence built as it follows in comments.

Original entry on oeis.org

9, 2, 45, 2, 11, 2, 225, 2, 11, 2, 56, 2, 11, 2, 1125, 2, 11, 2, 56, 2, 11, 281, 2, 11, 2, 56, 2, 11, 2, 5625

Offset: 0

Richard Choulet, Mar 30 2010

If we denote "A" the finite sequence between a(2^(n-1)-2) and a(2^n-2), the subsequence of a between

a(2^(n)-2) and a(2^(n+1)-2) is given by: " A - a(3*2^(n-1)-2) - A" for every n>=2.

			a(4)=a(2*3-2)=(9*4-1)/4=11. a(14)=a(2^4-2)==9*5^3=125*9=1125.
Between a(2) and a(6) the subsequence is "2, 11, 2"; then between a(6) and a(14) the subsequence of a is:
"2, 11, 2, a(10)=56, 2, 11, 2".
It seems that this new sequence gives the number of 2 in the sets of 2 of the sequence A174835.

Cf. A174835.

a(2n+1)=2. a(2^(n+1)-2)=9*5^n. a(3*2^n-2)=(9*5^n-1)/4.

A176753 Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=1, k=0 and l=-2.

Original entry on oeis.org

1, 1, 0, -1, -4, -12, -34, -93, -248, -644, -1622, -3932, -9054, -19314, -36066, -48953, 8372, 415848, 2180870, 8609676, 29858358, 95443242, 286747530, 815867808, 2199049782, 5577559986, 13083598882, 27240793594, 44583397354

Offset: 0

Richard Choulet, Apr 25 2010

			a(2)=2*1*1-2=0. a(3)=1-2=-1. a(4)=2*1*(-1)-2=-4.

Cf. A176752.

l:=-2: : k := 0 : m:=1:d(0):=1:d(1):=m: for n from 1 to 30 do d(n+1):=sum(d(p)*d(n-p)+k, p=0..n)+l:od :
taylor((1-sqrt(1-4*z*(d(0)-z*d(0)^2+z*m+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z), z=0, 30); seq(d(n), n=0..30);

G.f f: f(z)=(1-sqrt(1-4*z*(a(0)-z*a(0)^2+z*a(1)+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z) (k=0, l=-2).

Conjecture: (n+1)*a(n) +2*(1-3*n)*a(n-1) +(9*n-13)*a(n-2) +2*(2*n-9)*a(n-3) +8*(4-n)*a(n-4)=0. - R. J. Mathar, Jul 24 2012

A176755 Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=3, k=0 and l=-2.

Original entry on oeis.org

1, 3, 4, 15, 52, 208, 846, 3579, 15456, 68096, 304570, 1379980, 6319978, 29211278, 136086710, 638364319, 3012609980, 14293438828, 68139158918, 326218902372, 1567802352910, 7561126873098, 36581288824402, 177496766695528

Offset: 0

Richard Choulet, Apr 25 2010

			a(2)=2*1*3-2=4. a(3)=2*1*4+3^2-2=15. a(4)=2*1*15+2*3*4-2=52.

Cf. A176654.

l:=-2: : k := 0 : m:=3:d(0):=1:d(1):=m: for n from 1 to 30 do d(n+1):=sum(d(p)*d(n-p)+k, p=0..n)+l:od :
taylor((1-sqrt(1-4*z*(d(0)-z*d(0)^2+z*m+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z), z=0, 30); seq(d(n), n=0..30);

G.f f: f(z)=(1-sqrt(1-4*z*(a(0)-z*a(0)^2+z*a(1)+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z) (k=0, l=-2).

Conjecture: (n+1)*a(n) +2*(1-3*n)*a(n-1) +(n+3)*a(n-2) +2*(10*n-33)*a(n-3) +16*(4-n)*a(n-4) =0. - R. J. Mathar, Jul 24 2012

A177126 Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=10, k=1 and l=1.

Original entry on oeis.org

1, 10, 23, 150, 765, 5065, 32337, 223672, 1556583, 11178843, 81228819, 599868763, 4475307567, 33731219901, 256268778463, 1961208117130, 15101975890677, 116936866669157, 909887821312929, 7110983852617913, 55793178281433653

Offset: 0

Richard Choulet, May 03 2010

			a(2)=2*10+2+1=23. a(3)=2*1*23+2+10^2+1+1=150.

Cf. A177125.

l:=1: : k := 1 : m :=10: d(0):=1:d(1):=m: for n from 1 to 32 do d(n+1):=sum(d(p)*d(n-p)+k, p=0..n)+l:od :
taylor((1-sqrt(1-4*z*(d(0)-z*d(0)^2+z*m+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z), z=0, 34); seq(d(n), n=0..32);

G.f f: f(z)=(1-sqrt(1-4*z*(a(0)-z*a(0)^2+z*a(1)+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z) (k=1, l=1).

Conjecture: n*(n+1)*a(n) -n*(7*n-2)*a(n-1) -3*n*(7*n-17)*a(n-2) +n*(83*n-250)*a(n-3) -84*n*(n-4)*a(n-4) +28*n*(n-5)*a(n-5) =0. - R. J. Mathar, Jul 24 2012

A177131 Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=10, k=0 and l=1.

Original entry on oeis.org

1, 10, 21, 143, 707, 4716, 29579, 203622, 1399099, 9961582, 71585287, 523465627, 3864076389, 28826865756, 216722056701, 1641392860951, 12507535829603, 95839985593950, 737953189846751, 5707113130311621, 44310704176742745

Offset: 0

Richard Choulet, May 03 2010

			a(2)=2*1*10+1=21. a(3)=2*1*21+100+1=143.

Cf. A177130.

l:=1: : k := 0 : m :=10: d(0):=1:d(1):=m: for n from 1 to 28 do d(n+1):=sum(d(p)*d(n-p)+k, p=0..n)+l:od :
taylor((1-sqrt(1-4*z*(d(0)-z*d(0)^2+z*m+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z), z=0, 31); seq(d(n), n=0..29);

G.f f: f(z)=(1-sqrt(1-4*z*(a(0)-z*a(0)^2+z*a(1)+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z) (k=0, l=1).

Conjecture: (n+1)*a(n) +2*(-3*n+1)*a(n-1) +(-27*n+59)*a(n-2) +64*(n-3)*a(n-3) +32*(-n+4)*a(n-4)=0. - R. J. Mathar, Jul 24 2012

A177175 Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=6, k=1 and l=-1.

Original entry on oeis.org

1, 6, 13, 64, 287, 1515, 8143, 46030, 265909, 1572193, 9443997, 57529101, 354394057, 2204333079, 13823770729, 87311462772, 554904606279, 3546103422655, 22772157825695, 146876986425311, 951065019090195

Offset: 0

Richard Choulet, May 04 2010

			a(2)=2*1*6+2-1=13. a(3)=2*1*13+36+2+1-1=64.

Cf. A176832.

l:=-1: : k := 1 : m:=6:d(0):=1:d(1):=m: for n from 1 to 30 do d(n+1):=sum(d(p)*d(n-p)+k, p=0..n)+l:od :
taylor((1-sqrt(1-4*z*(d(0)-z*d(0)^2+z*m+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z), z=0, 30); seq(d(n), n=0..30);

G.f f: f(z)=(1-sqrt(1-4*z*(a(0)-z*a(0)^2+z*a(1)+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z) (k=1, l=-1).

Conjecture: (n+1)*a(n) +(2-7*n)*a(n-1) +(19-5*n)*a(n-2) +(43*n-134)*a(n-3) +4*(53-13*n)*a(n-4) +20*(n-5)*a(n-5)=0. - R. J. Mathar, Jul 24 2012

A177180 Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=10, k=1 and l=-1.

Original entry on oeis.org

1, 10, 21, 144, 711, 4747, 29767, 205078, 1409645, 10043729, 72216773, 528438373, 3903255409, 29138576719, 219209569841, 1661343858524, 12668020020047, 97135000445375, 748428139988567, 5792032911677831, 45000447097568843

Offset: 0

Richard Choulet, May 04 2010

Cf. A177179.

l:=-1: : k := 1 : for m from 0 to 10 do d(0):=1:d(1):=m: for n from 1 to 30 do d(n+1):=sum(d(p)*d(n-p)+k,p=0..n)+l:od :
taylor((1-sqrt(1-4*z*(d(0)-z*d(0)^2+z*m+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z),z=0,30);seq(d(n),n=0..30): od;

G.f f: f(z)=(1-sqrt(1-4*z*(a(0)-z*a(0)^2+z*a(1)+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z) (k=1, l=-1).

Conjecture: (n+1)*a(n) +(2-7*n)*a(n-1) +3*(17-7*n)*a(n-2) +(91*n-278)*a(n-3) +4*(101-25*n)*a(n-4) +36*(n-5)*a(n-5)=0. - R. J. Mathar, Jul 24 2012

cp's OEIS Frontend

User: Richard Choulet

A185106 Column 4 of A181783.

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A181783 Array described in comments to A053482, here read by increasing antidiagonals. See comments below.

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A159035 a(0)=1=a(1), a(2)=2, a(3)=5; thereafter a(n+3)=4*a(n+2)-4*a(n+1)+2*a(n) for n>=1.

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A174837 Sequence built as it follows in comments.

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A176753 Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=1, k=0 and l=-2.

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A176755 Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=3, k=0 and l=-2.

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A177126 Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=10, k=1 and l=1.

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A177131 Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=10, k=0 and l=1.

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A177175 Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=6, k=1 and l=-1.

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A159035 a(0)=1=a(1), a(2)=2, a(3)=5; thereafter a(n+3)=4a(n+2)-4a(n+1)+2*a(n) for n>=1.