A097806 -id:A097806 - cp's OEIS frontend

A128380 A097806^24 * A000594.

1, 0, -48, -24, 1104, 1128, -15892, -25368, 156240, 360640, -1057908, -3600696, 4417678, 26438568, -3155508, -144207816, -112109568, 564538680, 1002957320, -1344487080, -5096138658, -111333800, 17182425012, 17552839368, -34668142443, -86942440944, 4993723500, 236551774320

Offset: 1

Gary W. Adamson, Feb 28 2007

sign

Conjecture: Given the infinite set of sequences generated from the pairwise operation on A000594 (A097806^k * A000594), k = 24, (A128380) is the only sequence in the set with a zero. The sequence generated from k=23 = (1, -1, -47, 23, 1081, 47, -15939, ...). Analogous conjecture with the partial sum operator: (Cf. A128378, A128379); in which zeros are conjectured to occur only with k=23 and k=24. A128380 mod 24 = 1, 0, 0, 0, 0, 0, -4, 0, 0, 16, ...

Cf. A000594, A097806, A128378, A128379.

Nest[Prepend[Most[#] + Rest[#], First[#]] &, RamanujanTau[Range[30]], 24] (* Amiram Eldar, Jan 08 2025 *)

Pairwise operation performed 24 times on A000594

More terms from Amiram Eldar, Jan 08 2025

A128540 Triangle A127647 * A097806, read by rows.

Original entry on oeis.org

1, 1, 1, 0, 2, 2, 0, 0, 3, 3, 0, 0, 0, 5, 5, 0, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 13, 13, 0, 0, 0, 0, 0, 0, 21, 21, 0, 0, 0, 0, 0, 0, 0, 34, 34, 0, 0, 0, 0, 0, 0, 0, 0, 55, 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 89, 89, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 144, 144

Offset: 1

Gary W. Adamson, Mar 10 2007

Row sums = A094895 starting (1, 2, 4, 6, 10, 16, 26, ...). A128541 = A097806 * A127647.

			First few rows of the triangle:
  1;
  1, 1;
  0, 2, 2;
  0, 0, 3, 3;
  0, 0, 0, 5, 5;
  0, 0, 0, 0, 8, 8;
  ...

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A127647, A097806, A128541.

[k eq n select Fibonacci(n) else k eq n-1 select Fibonacci(n) else 0: k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 11 2019

Table[If[k==n || k==n-1, Fibonacci[n], 0], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Jul 11 2019 *)

T(n,k) = if(k==n || k==n-1, fibonacci(n), 0); \\ G. C. Greubel, Jul 11 2019

def T(n, k):
    if (k==n): return fibonacci(n)
    elif (k==n-1): return fibonacci(n)
    else: return 0
[[T(n, k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Jul 11 2019

A127646 * A097806 as infinite lower triangular matrices.

A131108 T(n,k) = 2*A007318(n,k) - A097806(n,k).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 2, 6, 5, 1, 2, 8, 12, 7, 1, 2, 10, 20, 20, 9, 1, 2, 12, 30, 40, 30, 11, 1, 2, 14, 42, 70, 70, 42, 13, 1, 2, 16, 56, 112, 140, 112, 56, 15, 1, 2, 18, 72, 168, 252, 252, 168, 72, 17, 1, 2, 20, 90, 240, 420, 504, 420, 240, 90, 19, 1

Offset: 0

Gary W. Adamson, Jun 15 2007

Row sums give A095121.

Triangle T(n,k), 0 <= k <= n, read by rows given by [1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 18 2007

			First few rows of the triangle are:
  1;
  1,  1;
  2,  3,  1;
  2,  6,  5,  1;
  2,  8, 12,  7, 1;
  2, 10, 20, 20, 9, 1;
...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A007318, A095121, A097806.

function T(n,k)
  if k eq n-1 then return 2*n-1;
  elif k eq n then return 1;
  else return 2*Binomial(n,k);
  end if;
  return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 18 2019

seq(seq( `if`(k=n-1, 2*n-1, `if`(k=n, 1, 2*binomial(n,k))), k=0..n), n=0..12); # G. C. Greubel, Nov 18 2019

Table[If[k==n-1, 2*n-1, If[k==n, 1, 2*Binomial[n, k]]], {n,0,12}, {k,0, n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

T(n,k) = if(k==n-1, 2*n-1, if(k==n, 1, 2*binomial(n,k))); \\ G. C. Greubel, Nov 18 2019

@CachedFunction
def T(n, k):
    if (k==n-1): return 2*n-1
    elif (k==n): return 1
    else: return 2*binomial(n,k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 18 2019

Twice Pascal's triangle minus A097806, the pairwise operator.

G.f.: (1-x*y+x^2+x^2*y)/((-1+x+x*y)*(x*y-1)). - R. J. Mathar, Aug 11 2015

Corrected by Philippe Deléham, Dec 17 2007

More terms added and data corrected by G. C. Greubel, Nov 18 2019

A128541 Triangle, A097806 * A127647, read by rows.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 0, 0, 2, 3, 0, 0, 0, 3, 5, 0, 0, 0, 0, 5, 8, 0, 0, 0, 0, 0, 8, 13, 0, 0, 0, 0, 0, 0, 13, 21, 0, 0, 0, 0, 0, 0, 0, 21, 34, 0, 0, 0, 0, 0, 0, 0, 0, 34, 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55, 89, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 89, 144, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 144, 233

Offset: 0

Gary W. Adamson, Mar 10 2007

Row sums = A000045 starting (1, 2, 3, 5, 8, 13, ...). A128540 = A127647 * A097806.

			First few rows of the triangle:
  1;
  1, 1;
  0, 1, 2;
  0, 0, 2, 3;
  0, 0, 0, 3, 5;
  0, 0, 0, 0, 5, 8;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000045, A097806, A127647, A128540.

[k eq n select Fibonacci(n+1) else k eq n-1 select Fibonacci(n) else 0: k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 11 2019

Table[If[k==n, Fibonacci[n+1], If[k==n-1, Fibonacci[n], 0]], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 11 2019 *)

T(n,k) = if(k==n, fibonacci(n+1), if(k==n-1, fibonacci(n), 0)); \\ G. C. Greubel, Jul 11 2019

def T(n, k):
    if (k==n): return fibonacci(n+1)
    elif (k==n-1): return fibonacci(n)
    else: return 0
[[T(n, k) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Jul 11 2019

A097806 * A127647 as infinite lower triangular matrices.

More terms added by G. C. Greubel, Jul 11 2019

A129573 A097806 * A129372.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Gary W. Adamson, Apr 22 2007

			First few rows of the triangle are:
  1;
  1, 1;
  1, 1, 1;
  1, 0, 1, 1;
  1, 0, 0, 1, 1;
  1, 1, 0, 0, 1, 1;
  1, 1, 0, 0, 0, 1, 1;
  1, 0, 0, 0, 0, 0, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A129574.

Cf. A129572, A129372, A097806.

T(n,k)=(n%k==0 && n/k%2) || ((n-1)%k==0 && (n-1)/k%2) \\ Andrew Howroyd, Aug 10 2018

A097806 * A129372 as infinite lower triangular matrices.

Duplicate a(35) removed and terms a(56) and beyond from Andrew Howroyd, Aug 10 2018

A131033 A130296 * A097806.

Original entry on oeis.org

1, 3, 1, 4, 2, 1, 5, 2, 2, 1, 6, 2, 2, 2, 1, 7, 2, 2, 2, 2, 1, 8, 2, 2, 2, 2, 2, 1, 9, 2, 2, 2, 2, 2, 2, 1, 10, 2, 2, 2, 2, 2, 2, 2, 1, 11, 2, 2, 2, 2, 2, 2, 2, 2, 1, 12, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 13, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 14, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1

Offset: 1

Gary W. Adamson, Jun 10 2007

Row sums give A016777.

A131032 = A097806 * A130296. [corrected by Georg Fischer, Oct 10 2021]

			First few rows of the triangle are:
1;
3, 1;
4, 2, 1;
5, 2, 2, 1;
6, 2, 2, 2, 1;
7, 2, 2, 2, 2, 1;
...

Cf. A016777, A097806, A130296.

A130296 * A097806 as infinite lower triangular matrices; where A130296 = (1; 2,1; 3,1,1;...) and A097806 = the pairwise operator.

Definition corrected, a(36)=2 inserted and more terms from Georg Fischer, Oct 10 2021

More terms from Michel Marcus, Oct 11 2021

A133599 A097806 * A133080 * A007318.

Original entry on oeis.org

1, 3, 1, 3, 3, 1, 3, 7, 5, 1, 3, 9, 10, 5, 1, 3, 13, 22, 18, 7, 1, 3, 15, 31, 34, 21, 7, 1, 3, 19, 51, 75, 65, 33, 9, 1, 3, 21, 64, 111, 120, 83, 36, 9, 1, 3, 25, 92, 196, 266, 238, 140, 52, 11, 1

Offset: 1

Gary W. Adamson, Sep 18 2007

Row sums = A133600: (1, 4, 7, 16, 28, 64, 112, ...).

			First few rows of the triangle:
  1;
  3,  1;
  3,  3,  1;
  3,  7,  5,  1;
  3,  9, 10,  5,  1;
  3, 13, 22, 18,  7,  1;
  ...

Cf. A097806, A133080, A133600.

A133599aux := proc(n,k)
    add(A097806(n,j)*A133080(j,k),j=k..n) ;
end proc:
A133599 := proc(n,k)
    add(A133599aux(n,j)*binomial(j-1,k-1),j=k..n) ;
end proc: # R. J. Mathar, Jun 20 2015

A097806 * A133080 * A007318 as infinite lower triangular matrices, where A097806 = the pairwise operator and A133080 = an interpolation operator.

A134315 A134309 * A097806.

Original entry on oeis.org

1, 1, 1, 0, 2, 2, 0, 0, 4, 4, 0, 0, 0, 8, 8, 0, 0, 0, 0, 16, 16, 0, 0, 0, 0, 0, 32, 32, 0, 0, 0, 0, 0, 0, 64, 64

Offset: 1

Gary W. Adamson, Oct 19 2007

A134315 * [1,2,3,...] = A128135: (1, 3, 10 28, 72, 176, 416, ...).

Triangle read by rows given by [1,-1,0,0,0,0,0,0,...] DELTA [1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 20 2007

			First few rows of the triangle are:
  1;
  1, 1;
  0, 2, 2;
  0, 0, 4, 4;
  0, 0, 0, 8, 8;
  ...

Cf. A134309, A097806, A128135.

A134309 * A134315 as infinite lower triangular matrices. Triangle read by rows, for n>1, (n-1) zeros followed by 2^(n-1), 2^(n-1). As an infinite lower triangular matrix, (1, 1, 2, 4, 8, ...) in the main diagonal and (1, 2, 4, 8, ...) in the subdiagonal.

G.f.: (-1-x+x*y)/(-1+2*x*y). - R. J. Mathar, Aug 11 2015

A129479 Triangle read by rows: A054523 * A097806 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 4, 0, 0, 1, 1, 4, 3, 1, 0, 1, 1, 6, 0, 0, 0, 0, 1, 1, 6, 2, 1, 1, 0, 0, 1, 1, 6, 2, 2, 0, 0, 0, 0, 1, 1, 8, 4, 0, 1, 1, 0, 0, 0, 1, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 4, 4, 2, 1, 1, 0, 0, 0, 0, 1, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Gary W. Adamson, Apr 17 2007

			First few rows of the triangle:
  1;
  2, 1;
  2, 1, 1;
  3, 1, 1, 1;
  4, 0, 0, 1, 1;
  4, 3, 1, 0, 1, 1;
  6, 0, 0, 0, 0, 1, 1;
  6, 2, 1, 1, 0, 0, 1, 1;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000010 (alternating row sums), A053158 (row sums).
Cf. A000038, A019590, A054523, A054977.
Cf. A063524, A097806, A126246, A183918.

A054523:= func< n,k | n eq 1 select 1 else (n mod k) eq 0 select EulerPhi(Floor(n/k)) else 0 >;
A129479:= func< n,k | k le n-1 select A054523(n,k) + A054523(n,k+1) else 1 >;
[A129479(n,k): k in [1..n], n in [1..16]]; // G. C. Greubel, Feb 11 2024

A054523[n_, k_]:= If[n==1, 1, If[Divisible[n,k], EulerPhi[n/k], 0]];
T[n_, k_]:= If[kA054523[n, j+k], {j,0,1}], 1];
Table[T[n,k],{n,16},{k,n}]//Flatten (* G. C. Greubel, Feb 11 2024 *)

def A054523(n,k):
    if (k==n): return 1
    elif (n%k): return 0
    else: return euler_phi(n//k)
def A129479(n, k):
    if k<0 or k>n: return 0
    elif k==n: return 1
    else: return A054523(n,k) + A054523(n,k+1)
flatten([[A129479(n, k) for k in range(1,n+1)] for n in range(1,17)]) # G. C. Greubel, Feb 11 2024

Sum_{k=1..n} T(n, k) = A053158(n) (row sums).
T(n, 1) = A126246(n).
From G. C. Greubel, Feb 11 2024: (Start)
T(n, k) = A054523(n, k) + A054523(n, k+1) for k < n, otherwise 1.
T(2*n-1, n) = A019590(n).
T(2*n, n) = A054977(n).
T(2*n+1, n) = A000038(n).
T(3*n, n) = A063524(n-1).
T(3*n-2, n) = A183918(n+2).
Sum_{k=1..n} (-1)^(k-1) * T(n, k) = A000010(n). (End)

A130452 Triangle read by rows: A097806 * A130321 as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 12, 6, 3, 1, 24, 12, 6, 3, 1, 48, 24, 12, 6, 3, 1, 96, 48, 24, 12, 6, 3, 1, 192, 96, 48, 24, 12, 6, 3, 1, 384, 192, 96, 48, 24, 12, 6, 3, 1, 768, 384, 192, 96, 48, 24, 12, 6, 3, 1, 1536, 768, 384, 192, 96, 48, 24, 12, 6, 3, 1, 3072, 1536, 768, 384, 192, 96, 48, 24, 12, 6, 3, 1

Offset: 1

Gary W. Adamson, May 26 2007

Row sums = A033484: (1, 4, 10, 22, 46, 94, 190, ...).

			First few rows of the triangle:
   1;
   3,  1;
   6,  3,  1;
  12,  6,  3,  1;
  24, 12,  6,  3,  1;
  48, 24, 12,  6,  3,  1;
  ...

Cf. A003945, A033484, A097806, A130321.

A097806 * A130321 as infinite lower triangular matrices. A097806 = the pairwise operator, A130321 = [1; 2,1; 4,2,1; ...]. Triangle, A003945 (1, 3, 6, 12, 24, 48, ...) in every column.

a(28) = 1 inserted and more terms from Georg Fischer, May 29 2023

cp's OEIS Frontend

A128380 A097806^24 * A000594.

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A128540 Triangle A127647 * A097806, read by rows.

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A131108 T(n,k) = 2*A007318(n,k) - A097806(n,k).

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A128541 Triangle, A097806 * A127647, read by rows.

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A129573 A097806 * A129372.

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A131033 A130296 * A097806.

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A133599 A097806 * A133080 * A007318.

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