A171647 -id:A171647 - cp's OEIS frontend

A123148 Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial p(n,x) defined by p(0,x) = -1, p(1,x) = x-2, p(n,x) = -xp(n-1,x) + 2p(n-2,x) for n >= 3 and 0 <= k <= n.

-1, -2, 1, -2, 2, -1, -4, 4, -2, 1, -4, 8, -6, 2, -1, -8, 12, -12, 8, -2, 1, -8, 24, -24, 16, -10, 2, -1, -16, 32, -48, 40, -20, 12, -2, 1, -16, 64, -80, 80, -60, 24, -14, 2, -1, -32, 80, -160, 160, -120, 84, -28, 16, -2, 1, -32, 160, -240, 320, -280, 168, -112, 32, -18, 2, -1

Offset: 0

Roger L. Bagula, Oct 01 2006

			The first few polynomials, p(n,x), are:
  p(0,x) = -1;
  p(1,x) = -2 +   x;
  p(2,x) = -2 + 2*x -   x^2;
  p(3,x) = -4 + 4*x - 2*x^2 +   x^3;
  p(4,x) = -4 + 8*x - 6*x^2 + 2*x^3 - x^4;
The triangle, T(n, k) = [x^k] p(n, x), begins as:
  -1;
  -2,  1;
  -2,  2,  -1;
  -4,  4,  -2,  1;
  -4,  8,  -6,  2,  -1;
  -8, 12, -12,  8,  -2,  1;
  -8, 24, -24, 16, -10,  2, -1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000034, A001045, A016116, A038754, A046854, A171647.

A123148:=func< n,k | (-1)^(k+1)*2^Floor((n-k+1)/2)*Binomial( Floor((n+k)/2), k) >;
[A123148(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 17 2023

p[0]:=-1: p[1]:=x-2: for n from 2 to 10 do p[n]:=sort(expand(-x*p[n-1]+2*p[n-2])) od: for n from 0 to 10 do seq(coeff(p[n],x,k),k=0..n) od; # yields sequence in triangular form

p[0,x]= -1; p[1,x]= x-2; p[k_, x_]:= p[k,x]= -x*p[k-1,x] + 2*p[k-2,x];
T[n_, k_]:= Coefficient[p[n, x], x, k];
Table[T[n,k], {n,0,12},{k,0,n}]//Flatten

def A123148(n,k): return (-1)^(k+1)*2^((n-k+1)//2)*binomial((n+k)//2, k)
flatten([[A123148(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 17 2023

T(n, k) = [x^k]( p(n,x) ), where p(0,x) = -1, p(1,x) = x-2, p(n,x) = -x*p(n-1,x) + 2*p(n-2,x).
Sum_{k=0..n} T(n, k) = -1.
Sum_{k=0..n} (-1)^k * T(n,k) = -A001045(n+2).
From G. C. Greubel, Jul 17 2023: (Start)
T(n,k) = (-1)^(k+1)*2^Floor((n-k+1)/2)*Binomial( Floor((n+k)/2), k).
T(n,k) = (-1)^(k+1)*2^Floor((n-k+1)/2)*A046854(n,k).
T(n,0) = -A016116(n+1).
T(n,1) = A171647(n).
Sum_{k=0..n} (-1)^k * abs(T(n,k)) = 1.
Sum_{k=0..floor(n/2)} T(n-k,k) = - A000034(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k,k) = -A038754(n). (End)

Edited by N. J. A. Sloane, Oct 29 2006

A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

Original entry on oeis.org

1, 1, 3, 17, 155, 2677, 73327, 3578339, 329652351

Offset: 1

Thomas Scheuerle, Aug 19 2022

This sequence can be calculated by a recursive algorithm:
Let B1 be an array of finite length, the "1" denotes that it is the first generation. Let B1' be the reversed version of B1. Let C be the element-wise product C = B1 * B1'. Then B2 is a concatenation of taking each element of B1 and add all divisors of the corresponding element in C. If we start with B1 = {1} then we get this sequence of arrays: B2 = {2}, B3 = {3, 4, 6}, ... . a(n) is the length of the array Bn. In short the length of Bn+1 and so a(n+1) is the sum over A000005(Bn * Bn').
The transform used in the definition of this sequence is its own inverse, so if c = S(b) then b = S(c). The eigensequence is 2^n = S(2^n).
There exist some transformation pairs of infinite sequences in the database:
  A000027 <--> A000142; A000079 <--> A000079; A000045 <--> A001654;
  A026549 <--> A038754; A100071 <--> A001405; A058295 <--> A------;
  A111286 <--> A098011; A093968 <--> A205825; A166447 <--> A------;
  A098558 <--> A004277; A010551 <--> A087811; A100538 <--> A329227;
  A079352 <--> A------; A082458 <--> A------; A008233 <--> A264635;
  A138278 <--> A------; A006501 <--> A264557; A336496 <--> A------;
  A019464 <--> A------; A062112 <--> A------; A171647 <--> A359039;
  A279312 <--> A------; A031923 <--> A------.
These transformation pairs are conjectured:
  A137326 <--> A------; A066332 <--> A300902; A208147 <--> A308546;
  A057895 <--> A------; A349080 <--> A------; A019442 <--> A------;
  A349079 <--> A------.
("A------" means not yet in the database.)
Some sequences in the lists above may need offset adjustment to force a beginning with 1,2,... in the transformation.
If we allowed signed rational numbers, further interesting transformation pairs could be observed. For example, 1/n will transform into factorials with alternating sign. 2^(-n) transforms into ones with alternating sign and 1/A000045(n) into A000045 with alternating sign.

			a(4) = 17. The 17 transformation pairs of length 4 are:
  {1, 2, 3, 4}  = S({1, 2, 6, 24}).
  {1, 2, 3, 5}  = S({1, 2, 6, 15}).
  {1, 2, 3, 6}  = S({1, 2, 6, 12}).
  {1, 2, 3, 9}  = S({1, 2, 6, 9}).
  {1, 2, 3, 12} = S({1, 2, 6, 8}).
  {1, 2, 3, 21} = S({1, 2, 6, 7}).
  {1, 2, 4, 5}  = S({1, 2, 4, 20}).
  {1, 2, 4, 6}  = S({1, 2, 4, 12}).
  {1, 2, 4, 8}  = S({1, 2, 4, 8}).
  {1, 2, 4, 12} = S({1, 2, 4, 6}).
  {1, 2, 4, 20} = S({1, 2, 4, 5}).
  {1, 2, 6, 7}  = S({1, 2, 3, 21}).
  {1, 2, 6, 8}  = S({1, 2, 3, 12}).
  {1, 2, 6, 9}  = S({1, 2, 3, 9}).
  {1, 2, 6, 12} = S({1, 2, 3, 6}).
  {1, 2, 6, 15} = S({1, 2, 3, 5}).
  {1, 2, 6, 24} = S({1, 2, 3, 4}).
b(1) = 1 by definition, b(2) = 1+1 as 1 has only 1 as divisor.
a(3) = A000005(b(2)*b(2)) = 3.
The divisors of b(2) are 1,2,4. So b(3) can be b(2)+1, b(2)+2 and b(2)+4.
a(4) = A000005((b(2)+1)*(b(2)+4)) + A000005((b(2)+2)*(b(2)+2)) + A000005((b(2)+4)*(b(2)+1)) = 17.

Cf. A000005.
Cf. A000027, A000045, A000079, A000142, A001405.
Cf. A001654, A004277, A006501, A008233, A019442.
Cf. A010551, A019464, A026549, A031923, A038754.
Cf. A057895, A058295, A062112, A066332, A100071.
Cf. A079352, A082458, A087811, A093968, A098011.
Cf. A098558, A100538, A111286, A166447, A137326.
Cf. A138278, A171647, A205825, A208147, A264557.
Cf. A264635, A308546, A329227, A336496, A349079.
Cf. A349080, A359039.

cp's OEIS Frontend

A123148 Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial p(n,x) defined by p(0,x) = -1, p(1,x) = x-2, p(n,x) = -xp(n-1,x) + 2p(n-2,x) for n >= 3 and 0 <= k <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

Extensions

A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

Views

Author

Keywords

Comments

Examples

Crossrefs

cp's OEIS Frontend

A123148 Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial p(n,x) defined by p(0,x) = -1, p(1,x) = x-2, p(n,x) = -x*p(n-1,x) + 2*p(n-2,x) for n >= 3 and 0 <= k <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

Extensions

A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

Views

Author

Keywords

Comments

Examples

Crossrefs

A123148 Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial p(n,x) defined by p(0,x) = -1, p(1,x) = x-2, p(n,x) = -xp(n-1,x) + 2p(n-2,x) for n >= 3 and 0 <= k <= n.