A127648 -id:A127648 - cp's OEIS frontend

A132813 Triangle read by rows: A001263 * A127648 as infinite lower triangular matrices.

1, 1, 2, 1, 6, 3, 1, 12, 18, 4, 1, 20, 60, 40, 5, 1, 30, 150, 200, 75, 6, 1, 42, 315, 700, 525, 126, 7, 1, 56, 588, 1960, 2450, 1176, 196, 8, 1, 72, 1008, 4704, 8820, 7056, 2352, 288, 9, 1, 90, 1620, 10080, 26460, 31752, 17640, 4320, 405, 10

Offset: 0

Gary W. Adamson, Sep 01 2007

Also T(n,k) = binomial(n-1, k-1)*binomial(n, k-1), related to Narayana polynomials (see Sulanke reference). - Roger L. Bagula, Apr 09 2008

h-vector for cluster complex associated to the root system B_n. See p. 8, Athanasiadis and C. Savvidou. - Tom Copeland, Oct 19 2014

			First few rows of the triangle are:
  1;
  1,  2;
  1,  6,   3;
  1, 12,  18,   4;
  1, 20,  60,  40,   5;
  1, 30, 150, 200,  75,   6;
  1, 42, 315, 700, 525, 126, 7;
  ...

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
N. Alexeev and A. Tikhomirov, Singular Values Distribution of Squares of Elliptic Random Matrices and type-B Narayana Polynomials, arXiv preprint arXiv:1501.04615 [math.PR], 2015.
C. Athanasiadis and C. Savvidou, The local h-vector of the cluster subdivision of a simplex, arXiv preprint arXiv:1204.0362 [math.CO], 2012.
Robert. A. Sulanke, Counting Lattice Paths by Narayana Polynomials Electronic J. Combinatorics 7, No. 1, R40, 1-9, 2000.

Family of polynomials (see A062145): A008459 (c=1), this sequence (c=2), A062196 (c=3), A062145 (c=4), A062264 (c=5), A062190 (c=6).
Columns: A000012 (k=0), A002378 (k=1), A006011 (k=2), 4*A006542 (k=3), 5*A006857 (k=4), 6*A108679 (k=5), 7*A134288 (k=6), 8*A134289 (k=7), 9*A134290 (k=8), 10*A134291 (k=9).
Diagonals: A000027 (k=n), A002411 (k=n-1), A004302 (k=n-2), A108647 (k=n-3), A134287 (k=n-4).
Main diagonal: A000894.
Sums: (-1)^floor((n+1)/2)*A001405 (signed row), A001700 (row), A203611 (diagonal).
Cf. A001263, A007318, A127648, A281260.
Cf. A103371 (mirrored).

Flat(List([0..10],n->List([0..n], k->(k+1)*Binomial(n+1,k+1)*Binomial(n+1,k)/(n+1)))); # Muniru A Asiru, Feb 26 2019

a132813 n k = a132813_tabl !! n !! k
a132813_row n = a132813_tabl !! n
a132813_tabl = zipWith (zipWith (*)) a007318_tabl $ tail a007318_tabl
-- Reinhard Zumkeller, Apr 04 2014

/* triangle */ [[(k+1)*Binomial(n+1,k+1)*Binomial(n+1,k)/(n+1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 19 2014

P := (n, x) -> hypergeom([1-n, -n], [1], x): for n from 1 to 9 do PolynomialTools:-CoefficientList(simplify(P(n,x)),x) od; # Peter Luschny, Nov 26 2014

T[n_,k_]=Binomial[n-1,k-1]*Binomial[n,k-1]; Table[Table[T[n,k],{k,1,n}],{n,1,11}]; Flatten[%] (* Roger L. Bagula, Apr 09 2008 *)
P[n_, x_] := HypergeometricPFQ[{1-n, -n}, {1}, x]; Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten (* Jean-François Alcover, Nov 27 2014, after Peter Luschny *)

tabl(nn) = {for (n = 1, nn, for (k = 1, n, print1(binomial(n-1, k-1)*binomial(n, k-1) , ", ");););} \\ Michel Marcus, Feb 12 2014

def A132813(n,k): return binomial(n,k)*binomial(n+1,k)
print(flatten([[A132813(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 12 2025

T(n,k) = (k+1)*binomial(n+1,k+1)*binomial(n+1,k)/(n+1), n >= k >= 0.
From Roger L. Bagula, May 14 2010: (Start)
T(n, m) = coefficients(p(x,n)), where
p(x,n) = (1-x)^(2*n)*Sum_{k >= 0} binomial(k+n-1, k)*binomial(n+k, k)*x^k,
 or p(x,n) = (1-x)^(2*n)*Hypergeometric2F1([n, n+1], [1], x). (End)
T(n,k) = binomial(n,k) * binomial(n+1,k). - Reinhard Zumkeller, Apr 04 2014
These are the coefficients of the polynomials Hypergeometric2F1([1-n,-n], [1], x). - Peter Luschny, Nov 26 2014
G.f.: A(x,y) = A281260(x,y)/(1-A281260(x,y))/x. - Vladimir Kruchinin, Oct 10 2020

A134577 A127170 * A127648.

Original entry on oeis.org

1, 2, 2, 2, 0, 3, 3, 4, 0, 4, 2, 0, 0, 0, 5, 4, 4, 6, 0, 0, 6, 2, 0, 0, 0, 0, 0, 7, 4, 6, 0, 8, 0, 0, 0, 8, 3, 0, 6, 0, 0, 0, 0, 0, 9, 4, 4, 0, 0, 10, 0, 0, 0, 0, 10

Offset: 1

Gary W. Adamson, Nov 02 2007

Row sums = A007429: (1, 4, 5, 11, 7, 20, 9, 26, ...).
Left border = A000005: (1, 2, 2, 3, 2, 4, 2, ...).
A134577 * [1/1, 1/2, 1/3, ...] = A007425: (1, 3, 3, 6, 3, 9, 3, 10, ...).
A134577 * [1, 2, 3, ...] = A007433: (1, 6, 11, 27, 27, 66, ...).
A134577 * A000005 = A034761: (1, 6, 8, 23, 12, 48, ...).

			First few rows of the triangle:
  1;
  2, 2;
  2, 0, 3;
  3, 4, 0, 4;
  2, 0, 0, 0, 5
  4, 4, 6, 0, 0, 6;
  2, 0, 0, 0, 0, 0, 7;
  4, 6, 0, 8, 0, 0, 0, 8;
  ...

Cf. A051731, A127170, A127648, A127093, A007429, A127170, A007433, A034761.

A127170 * A127648 = A051731^2 * A127648 = A051731 * A127093.

A127649 A127648 * A054523 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 2, 6, 0, 3, 8, 4, 0, 4, 20, 0, 0, 0, 5, 12, 12, 6, 0, 0, 6, 42, 0, 0, 0, 0, 0, 7, 32, 16, 0, 8, 0, 0, 0, 8, 54, 0, 18, 0, 0, 0, 0, 0, 9, 40, 40, 0, 0, 10, 0, 0, 0, 0, 10, 110, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 48, 24, 24, 24, 0, 12, 0, 0, 0, 0, 0, 12, 156, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 84

Offset: 1

Gary W. Adamson, Jan 22 2007

Natural number transform of A054523.

Row sums = n^2, left column = A002618

			First few rows of the triangle are:
1;
2, 2;
6, 0, 3;
8, 4, 0, 4;
20, 0, 0, 0, 5;
12, 12, 6, 0, 0, 6;
42, 0, 0, 0, 0, 0, 7;
...

Cf. A127648, A002618, A054523.

A054523 := proc(n,k) if n mod k = 0 then numtheory[phi](n/k) ; else 0 ; fi ; end: A127649 := proc(n,k) A054523(n,k)*n ; end: for n from 1 to 20 do for k from 1 to n do printf("%d,",A127649(n,k)) ; od: od: # R. J. Mathar, Nov 01 2007

T(n,k)=n*A054523(n,k). - R. J. Mathar, Nov 01 2007

T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x - y. - Mats Granvik, Oct 08 2023

More terms from R. J. Mathar, Nov 01 2007

A127733 Square of A127648 = Triangle read by rows, n^2 preceded by (n-1) zeros.

Original entry on oeis.org

1, 0, 4, 0, 0, 9, 0, 0, 0, 16, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 49

Offset: 1

Gary W. Adamson, Jan 26 2007

			First few rows of the triangle are:
1;
0, 4;
0, 0, 9;
0, 0, 0, 16;
0, 0, 0, 0, 25;
...

Cf. A127648.

A127448 Triangle T(n,k) read by rows: matrix product A054525 * A127648.

Original entry on oeis.org

1, -1, 2, -1, 0, 3, 0, -2, 0, 4, -1, 0, 0, 0, 5, 1, -2, -3, 0, 0, 6, -1, 0, 0, 0, 0, 0, 7, 0, 0, 0, -4, 0, 0, 0, 8, 0, 0, -3, 0, 0, 0, 0, 0, 9, 1, -2, 0, 0, -5, 0, 0, 0, 0, 10, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 2, 0, -4, 0, -6, 0, 0, 0, 0, 0, 12, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 1, -2, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0

Offset: 1

Gary W. Adamson, Jan 14 2007

			First few rows of the triangle are;
1;
-1, 2;
-1, 0, 3;
0, -2, 0, 4;
-1, 0, 0, 0, 5;
1, -2, -3, 0, 0, 6;
-1, 0, 0, 0, 0, 0, 7;
0, 0, 0, -4, 0, 0, 0, 8;
0, 0, -3, 0, 0, 0, 0, 0, 9;
1, -2, 0, 0,-5, 0, 0, 0, 0, 10;
...

Cf. A000010, A008683, A051731.

A127648 := proc(n,k) if n = k then n; else 0 ; fi; end:
A054525 := proc(n,k) if k = n then 1; elif n mod k = 0 then numtheory[mobius](n/k) ; else 0 ; fi; end:
A127448 := proc(n,k) add( A054525(n,j)*A127648(j,k), j=k..n) ; end: seq(seq( A127448(n,k),k=1..n),n=1..15) ;

T(n,k) = sum _{j=k..n} A054525(n,j)*A127648(j,k) = k*A054525(n,k).
sum_{k=1..n} T(n,k) = A000010(n) (row sums).
T(n,1) = A008683(n).

Converted comments to formulas, extended - R. J. Mathar, Sep 11 2009
Corrected A-number typo in a formula - R. J. Mathar, Sep 17 2009
Corrected last example line by John Mason, Jan 07 2015

A128621 A127648 * A128174 as an infinite lower triangular matrix.

Original entry on oeis.org

1, 0, 2, 3, 0, 3, 0, 4, 0, 4, 5, 0, 5, 0, 5, 0, 6, 0, 6, 0, 6, 7, 0, 7, 0, 7, 0, 7, 0, 8, 0, 8, 0, 8, 0, 8, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 10, 0, 10, 0, 10, 0, 10, 0, 10, 11, 0, 11, 0, 11, 0, 11, 0, 11, 0, 11, 0, 12, 0, 12, 0, 12, 0, 12, 0, 12, 0, 12, 13, 0, 13, 0, 13, 0, 13, 0, 13, 0, 13, 0, 13

Offset: 1

Gary W. Adamson, Mar 14 2007

			First few rows of the triangle:
  1;
  0, 2;
  3, 0, 3;
  0, 4, 0, 4;
  5, 0, 5, 0, 5;
  ...

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

Cf. A000326, A123684, A127648, A128174.

Cf. A093005 (row sums).

[n*(1+(-1)^(n+k))/2: k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 13 2024

Table[n*(1+(-1)^(n+k))/2, {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 13 2024 *)

flatten([[n*(1+(-1)^(n+k))//2 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 13 2024

Odd rows: n terms of n, 0, n, ...; even rows, n terms of 0, n, 0, ...
T(n,k) = n if n+k even, T(n,k) = 0 if n+k odd.
Sum_{k=1..n} T(n, k) = A093005(n) (row sums).
From G. C. Greubel, Mar 13 2024: (Start)
T(n, k) = n*(1 + (-1)^(n+k))/2.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n+1)*A093005(n).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1-(-1)^n) * A000326(floor((n+1)/2)).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1 - (-1)^n)*A123684(floor((n+1)/2)). (End)

More terms added by G. C. Greubel, Mar 13 2024

A144824 Triangle read by rows, A054533 * A127648 (matrix product).

Original entry on oeis.org

1, -1, 2, -1, -2, 6, 0, -4, 0, 8, -1, -2, -3, -4, 20, 1, -2, -6, -4, 5, 12, -1, -2, -3, -4, -5, -6, 42, 0, 0, 0, -16, 0, 0, 0, 32, 0, 0, -9, 0, 0, -18, 0, 0, 54, 1, -2, 3, -4, -20, -6, 7, -8, 9, 40, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, 110

Offset: 1

Gary W. Adamson, Sep 21 2008

Row sums = A023896: (1, 1, 3, 4, 10, 16, 21, ...).
Right border = A002618: (1, 2, 6, 8, 20, 12, ...).
Left border = mu(n) = A008683 (n).

			Triangle A054533 starts as follows:
   1;
  -1,  1;
  -1, -1,  2;
   0, -2,  0,  2;
  -1, -1, -1, -1, 4;
   1, -1, -2, -1, 1, 2;
   ...
The first few rows of triangle A144824 are as follows:
   1;
  -1,  2;
  -1, -2,  6;
   0, -4,  0,  8;
  -1, -2, -3, -4, 20;
   1, -2, -6, -4,  5, 12;
  -1, -2, -3, -4, -5, -6, 42;
   ...

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A002260, A002618, A008683, A023896, A054533, A127648.

Triangle read by rows, A054533 * A127648 (matrix product). The operation is equivalent to taking termwise products of row A054533 terms and the natural numbers.
T(n, k) = k * Sum_{d|gcd(n,k)} d * mu(n/d) for n >= 1 and 1 <= k <= n. - Petros Hadjicostas, Jul 28 2019
a(n) = A002260(n)*A054533(n). - Jinyuan Wang, Jul 29 2019

A134464 (A127648 * A000012 + A000012 * A127773) - A000012.

Original entry on oeis.org

1, 2, 4, 3, 5, 8, 4, 6, 9, 13, 5, 7, 10, 14, 19, 6, 8, 11, 15, 20, 26, 7, 9, 12, 16, 21, 27, 34, 8, 10, 13, 17, 22, 28, 35, 43, 9, 11, 14, 18, 23, 29, 36, 44, 53, 10, 12, 15, 19, 24, 30, 37, 45, 54, 64

Offset: 1

Gary W. Adamson, Oct 26 2007

Row sums = A134465: (1, 6, 16, 32, 55, 86, ...).

			First few rows of the triangle:
  1;
  2,  4;
  3,  5,  8;
  4,  6,  9, 13;
  5,  7, 10, 14, 19;
  6,  8, 11, 15, 20, 26;
  7,  9, 12, 16, 21, 27, 34;
  ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A127648, A127773, A134465.

Flatten[Table[RecurrenceTable[{a[1]==i,a[n]==a[n-1]+n},a,{n,i}],{i,10}]] (* Harvey P. Dale, Nov 12 2013 *)

(A127648 * A000012 * A000012 * A127773) - A000012, as infinite lower triangular matrices.

A135223 Triangle A000012 * A127648 * A103451, read by rows.

Original entry on oeis.org

1, 3, 2, 6, 2, 3, 10, 2, 3, 4, 15, 2, 3, 4, 5, 21, 2, 3, 4, 5, 6, 28, 2, 3, 4, 5, 6, 7, 36, 2, 3, 4, 5, 6, 7, 8, 45, 2, 3, 4, 5, 6, 7, 8, 9, 55, 2, 3, 4, 5, 6, 7, 8, 9, 10, 66, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 78, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 91, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13

Offset: 1

Gary W. Adamson, Nov 23 2007

Row sums = A028387.

			First few rows of the triangle are:
   1;
   3, 2;
   6, 2, 3;
  10, 2, 3, 4;
  15, 2, 3, 4, 5;
...

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

Cf. A002260, A103451, A127648.

T:= function(n,k)
    if k=1 then return Binomial(n+1,2);
    else return k;
    fi; end;
Flat(List([1..15], n-> List([1..n], k-> T(n,k) ))); # G. C. Greubel, Nov 20 2019

[k eq 1 select Binomial(n+1,2) else k: k in [1..n], n in [1..15]]; // G. C. Greubel, Nov 20 2019

seq(seq( `if`(k=1, binomial(n+1,2), k), k=1..n), n=1..15); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k]= If[k==1, Binomial[n+1, 2], k]; Table[T[n, k], {n, 15}, {k,n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

T(n,k) = if(k==1, binomial(n+1,2), k); \\ G. C. Greubel, Nov 20 2019

@CachedFunction
def T(n,k):
    if (k==1): return binomial(n+1, 2)
    else: return k
[[T(n,k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Nov 20 2019

T(n,k) = A000012(n,k) * A127648(n,k) * A103451(n,k) as infinite lower triangular matrices. Replace left border of 1's in A002260 with (1, 3, 6, 10, 15, ...).

T(n, k) = k with T(n,1) = binomial(n+1, 2). - G. C. Greubel, Nov 20 2019

More terms added by G. C. Greubel, Nov 20 2019

A143256 Triangle read by rows, matrix multiplication A051731 * A128407 * A127648, 1<=k<=n.

Original entry on oeis.org

1, 1, -2, 1, 0, -3, 1, -2, 0, 0, 1, 0, 0, 0, -5, 1, -2, -3, 0, 0, 6, 1, 0, 0, 0, 0, 0, -7, 1, -2, 0, 0, 0, 0, 0, 0, 1, 0, -3, 0, 0, 0, 0, 0, 0, 1, -2, 0, 0, -5, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -11, 1, -2, -3, 0, 0, 6, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -13, 1, -2, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 14

Offset: 1

Gary W. Adamson, Aug 02 2008

Right border = n*mu(n) = A055615.

Row sums = A023900: (1, -1, -2, -1, -4, 2, -6,...).

			First few rows of the triangle =
1;
1, -2;
1, 0, -3;
1, -2, 0, 0;
1, 0, 0, 0, -5;
1, -2, -3, 0, 0, 6;
1, 0, 0, 0, 0, 0, -7;
...

Robert Israel, Table of n, a(n) for n = 1..9870

Cf. A008683, A055615, A051731, A128407, A127648, A023900.

seq(seq(`if`(i mod j = 0, j*numtheory:-mobius(j),0), j=1..i),i=1..20); # Robert Israel, Sep 07 2014

Table[If[Divisible[n, k], k MoebiusMu[k], 0], {n, 1, 14}, {k, 1, n}] (* Jean-François Alcover, Jun 19 2019 *)

A143256_row = lambda n: [k*moebius(k) if k.divides(n) else 0 for k in (1..n)]
for n in (1..10): print(A143256_row(n)) # Peter Luschny, Jan 05 2018

Triangle read by rows, A051731 * A128407 * A127648, 1<=k<=n

cp's OEIS Frontend

A132813 Triangle read by rows: A001263 * A127648 as infinite lower triangular matrices.

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A134577 A127170 * A127648.

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A127649 A127648 * A054523 as infinite lower triangular matrices.

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A127733 Square of A127648 = Triangle read by rows, n^2 preceded by (n-1) zeros.

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A127448 Triangle T(n,k) read by rows: matrix product A054525 * A127648.

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A128621 A127648 * A128174 as an infinite lower triangular matrix.

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A144824 Triangle read by rows, A054533 * A127648 (matrix product).

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A134464 (A127648 * A000012 + A000012 * A127773) - A000012.

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