A112742 -id:A112742 - cp's OEIS frontend

A144216 C(m,2)+C(n,2), m>=1, n>=1: a rectangular array R read by antidiagonals.

0, 1, 1, 3, 2, 3, 6, 4, 4, 6, 10, 7, 6, 7, 10, 15, 11, 9, 9, 11, 15, 21, 16, 13, 12, 13, 16, 21, 28, 22, 18, 16, 16, 18, 22, 28, 36, 29, 24, 21, 20, 21, 24, 29, 36, 45, 37, 31, 27, 25, 25, 27, 31, 37, 45, 55, 46, 39, 34, 31, 30, 31, 34, 39, 46, 55, 66, 56, 48, 42, 38, 36, 36, 38

Offset: 1

Clark Kimberling, Sep 14 2008

This is the accumulation array (as defined at A144112) of the weight array A144217.

As a triangular array read by rows (0; 1, 1; 3, 2, 3; 6, 4, 4, 6; ...), T(n,j) = (1/2)n(n+1-2j)+j(j-1) (1<=j<=n) is the sum of the distances from the vertex j of the path graph 1-2-...-n to all the other vertices. Example: T(4,2)=4 because in the path 1-2-3-4 the distances from vertex 2 to the vertices 1, 2, 3, 4 are 1, 0, 1, 2, respectively; 1+0+1+2=4.

			   0,  1,  3,  6, 10, 15, 21, 28, 36, 45, ...
   1,  2,  4,  7, 11, 16, 22, 29, 37, 46, ...
   3,  4,  6,  9, 13, 18, 24, 31, 39, 48, ...
   6,  7,  9, 12, 16, 21, 27, 34, 42, 51, ...
  10, 11, 13, 16, 20, 25, 31, 38, 46, 55, ...
  15, 16, 18, 21, 25, 30, 36, 43, 51, 60, ...
  21, 22, 24, 27, 31, 36, 42, 49, 57, 66, ...
  28, 29, 31, 34, 38, 43, 49, 56, 64, 73, ...
  36, 37, 39, 42, 46, 51, 57, 64, 72, 81, ...
  45, 46, 48, 51, 55, 60, 66, 73, 81, 90, ...
R(2,4) = binomial(2,2) + binomial(4,2) = 1 + 6 = 7.

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened).

Cf. A144217.

T := proc (n, j) if j <= n then (1/2)*n*(n+1-2*j)+j*(j-1) else 0 end if end proc: for n to 12 do seq(T(n, j), j = 1 .. n) end do; # yields sequence in triangular form

Table[n(n-m-1)+m(m+1)/2,{m,15},{n,m}] (* Paolo Xausa, Dec 21 2023 *)

R(m,n) = (m(m-1)+n(n-1))/2.

The sum of the terms in the upper left r X r submatrix is Sum_{n=1..r} Sum_{m=1..r} R(n,m) = A112742(r). - J. M. Bergot, Jun 18 2013

A210220 T(n, k) = -binomial(2n-k+2, k+1)hypergeom([2*n-k+3, 1], [k+2], 2). Triangle read by rows, T(n, k) for 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 12, 13, 4, 5, 20, 34, 24, 5, 6, 30, 70, 80, 40, 6, 7, 42, 125, 200, 166, 62, 7, 8, 56, 203, 420, 496, 314, 91, 8, 9, 72, 308, 784, 1211, 1106, 553, 128, 9, 10, 90, 444, 1344, 2576, 3108, 2269, 920, 174, 10, 11, 110, 615, 2160, 4956, 7476, 7274, 4352, 1461, 230, 11

Offset: 1

Clark Kimberling, Mar 19 2012

Previous name: Triangle of coefficients of polynomials v(n,x) jointly generated with A210217.

For a discussion and guide to related arrays, see A208510.

			First five rows:
  1
  2...2
  3...6....3
  4...12...13...4
  5...20...34...24...5
First three polynomials v(n,x): 1, 2 + 2x , 3 + 6x + 3x^2.

Cf. A210217, A210219, A208510, A016061, A112742.

T := (n,k) -> -binomial(2*n-k+2, k+1)*hypergeom([2*n-k+3, 1], [k+2], 2):
seq(seq(simplify(T(n,k)), k=1..n), n=1..10); # Peter Luschny, Oct 31 2019

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]     (* A210219 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]     (* A210220 *)
(* alternate program *)
T[n_,k_]:=Sum[Binomial[2*j+k-2,k-1],{j,1,n-k+1}];Flatten[Table[T[n,k],{n,1,11},{k,1,n}]] (* Detlef Meya, Dec 05 2023 *)

First and last term in row n: n.
Column 2: n*(n-1).
Column 3: A016061.
Column 4: A112742.
Row sums: -1+(even-indexed Fibonacci numbers).
Periodic alternating row sums: 1,0,0,1,0,0,1,0,0,...
u(n,x)=x*u(n-1,x)+v(n-1,x)+1,
v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
T(n,k) = Sum_{j=1..n-k+1} binomial(2*j+k-2,k-1). - Detlef Meya, Dec 05 2023

New name from Peter Luschny, Oct 31 2019

A350050 a(n) = (2n^4 - 6(-1)^nn^2 - 2n^2 + 3*(-1)^n - 3)/96.

Original entry on oeis.org

0, 0, 0, 2, 4, 14, 24, 52, 80, 140, 200, 310, 420, 602, 784, 1064, 1344, 1752, 2160, 2730, 3300, 4070, 4840, 5852, 6864, 8164, 9464, 11102, 12740, 14770, 16800, 19280, 21760, 24752, 27744, 31314, 34884, 39102, 43320, 48260, 53200, 58940, 64680, 71302, 77924, 85514

Offset: 0

Stefano Spezia, Dec 11 2021

Definitions: (Start)
The k-th exterior power of a vector space V of dimension n is a vector subspace spanned by elements, called k-vectors, that are the exterior product of k vectors v_i in V.
Given a square matrix A that describes the vectors v_i in terms of a basis of V, the k-th exterior power of the matrix A is the matrix that represents the k-vectors in terms of the basis of V. (End)
Conjectures: (Start)
For n > 1, a(n) is the absolute value of the trace of the 2nd exterior power of an n X n square matrix A(n) defined as A[i,j,n] = n - abs((n + 1)/2 - j) - abs((n + 1)/2 - i) (see A349107). Equivalently, a(n) is the absolute value of the coefficient of the term [x^(n-2)] in the characteristic polynomial of the matrix A(n), or the absolute value of the sum of all principal minors of A(n) of size 2.
For k > 2, the trace of the k-th exterior power of the matrix A(n) is equal to zero. (End)
The same conjectures hold for an n X n square matrix A(n) defined as A[i,j,n] = (n mod 2) + abs((n + 1)/2 - j) + abs((n + 1)/2 - i) (see A349108).

Winston de Greef, Table of n, a(n) for n = 0..10000
Wikipedia, Characteristic polynomial
Wikipedia, Exterior algebra
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A000982 (trace of matrix A(n)), A317614 (elements sum of matrix A(n)), A349107, A349108.

Cf. A006325, A112742, A338429.

Table[(2*n^4-6*(-1)^n*n^2-2*n^2+3*(-1)^n-3)/96,{n,0,45}]

a(n) = (2*n^4 - 6*(-1)^n*n^2 - 2*n^2 + 3*(-1)^n - 3)/96 \\ Winston de Greef, Jan 28 2024

O.g.f.: 2*x^3*(1 + x^2)/((1 - x)^5*(1 + x)^3).
E.g.f.: (x*(x^3 + 6*x^2 + 3*x + 3)*cosh(x) + (x^4 + 6*x^3 + 9*x^2 - 3*x - 3)*sinh(x))/48.
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8) for n > 7.
a(n) = A338429(n-2)/2 for n > 2.
a(2*n-1) = 2*A006325(n).
a(2*n) = A112742(n).
Sum_{n>2} 1/a(n) = (45 - 2*Pi^2 - 4*sqrt(3)*Pi*tanh(sqrt(3)*Pi/2))/4 = 0.920755957767250147865...

A208532 Mirror image of triangle in A125185; unsigned version of A120058.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 9, 10, 4, 5, 16, 28, 24, 8, 6, 25, 60, 80, 56, 16, 7, 36, 110, 200, 216, 128, 32, 8, 49, 182, 420, 616, 560, 288, 64, 9, 64, 280, 784, 1456, 1792, 1408, 640, 128, 10, 81, 408, 1344, 3024, 4704, 4992, 3456, 1408, 256

Offset: 0

Philippe Deléham, Feb 27 2012

Subtriangle of the triangle given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Equals A007318*A134309*A097806 as infinite lower triangular matrix.
Row sums are powers of 3 (A000244).
Diagonal sums are powers of 2 (A000079).

			Triangle begins :
1
2, 1
3, 4, 2
4, 9, 10, 4
5, 16, 28, 24, 8
6, 25, 60, 80, 56, 16
7, 36, 110, 200, 216, 128, 32
8, 49, 182, 420, 616, 560, 288, 64
9, 64, 280, 784, 1456, 1792, 1408, 640, 128
10, 81, 408, 1344, 3024, 4704, 4992, 3456, 1408, 256
Triangle (1, 1, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, ...) begins :
1
1, 0
2, 1, 0
3, 4, 2, 0
4, 9, 10, 4, 0
5, 16, 28, 24, 8, 0
6, 25, 60, 80, 56, 16, 0

Cf. Columns: A000027, A000290, A006331, A112742.

Cf. Diagonals: A011782, 2*A045623,

T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-1,k-1), T(0,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
G.f.: (1-y*x)/((1-x)*(1-(1+2*y)*x)).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A083085(n), A084567(n), A000012(n), A000027(n+1), A000244(n), A083065(n), A083076(n) for x = -3, -2, -1, 0, 1, 2, 3 respectively.

A233295 Riordan array ((1+x)/(1-x)^3, 2*x/(1-x)).

Original entry on oeis.org

1, 4, 2, 9, 10, 4, 16, 28, 24, 8, 25, 60, 80, 56, 16, 36, 110, 200, 216, 128, 32, 49, 182, 420, 616, 560, 288, 64, 64, 280, 784, 1456, 1792, 1408, 640, 128, 81, 408, 1344, 3024, 4704, 4992, 3456, 1408, 256, 100, 570, 2160, 5712, 10752, 14400, 13440, 8320, 3072, 512

Offset: 0

Philippe Deléham, Dec 07 2013

Subtriangle of the triangle in A208532.
Row sums are A060188(n+2).
Diagonal sums are A000295(n+2)=A125128(n+1)=A130103(n+2).

			Triangle begins :
1
4, 2
9, 10, 4
16, 28, 24, 8
25, 60, 80, 56, 16
36, 110, 200, 216, 128, 32
49, 182, 420, 616, 560, 288, 64
64, 280, 784, 1456, 1792, 1408, 640, 128
81, 408, 1344, 3024, 4704, 4992, 3456, 1408, 256
100, 570, 2160, 5712, 10752, 14400, 13440, 8320, 3072, 512

Cf. Columns: A000290, A006331, A112742.

Cf. Diagonal: A000079.

G.f. for the column k: 2^k*(1+x)/(1-x)^(k+3).
T(n,k) = 2^k*(binomial(n,k)+3*binomial(n,k+1)+2*binomial(n,k+2)), 0<=k<=n.
T(n,0) = 2*T(n-1,0)-T(n-2,0)+2, T(n,k)=2*T(n-1,k)+2*T(n-1,k-1)-2*T(n-2,k-1)-T(n-2,k) for k>=1, T(0,0)=1, T(1,0)=4, T(1,1)=2, T(n,k)=0 if k<0 or if k>n.
Sum_{k=0..n} T(n,k) = A060188(n+2).
Sum_{k=0..n} T(n,k)*(-1)^k = n+1.
T(n,k) = 2*sum_{j=1..n-k+1} T(n-j,k-1).
T(n,k) = 2^k*A125165(n,k).
T(n,n) = 2^n=A000079(n).
T(n,0) = (n+1)^2=A000290(n+1).
exp(2*x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(2*x)*(16 + 28*x + 24*x^2/2! + 8*x^3/3!) = 16 + 60*x + 200*x^2/2! + 616*x^3/3! + 1792*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), 2*x/(1 - x) ). Cf. A125165. - Peter Bala, Dec 21 2014

A303231 Total volume of all rectangular prisms with dimensions q, p+q and |q-p| such that p and q are prime, n = p+q and p < q.

Original entry on oeis.org

0, 0, 0, 0, 15, 0, 105, 80, 315, 280, 0, 168, 1287, 1232, 2145, 3136, 0, 2664, 4845, 6320, 6783, 11176, 0, 11088, 12075, 17888, 0, 14448, 0, 17640, 24273, 27776, 29667, 62560, 0, 61632, 0, 28272, 50505, 76720, 0, 99120, 68757, 141944, 79335, 163024, 0

Offset: 1

Wesley Ivan Hurt, Apr 20 2018

Index entries for sequences related to partitions

Cf. A010051, A028723, A112742.

Table[n*Sum[(n - i) (n - 2 i) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i] - PrimePi[n - i - 1]), {i, Floor[(n - 1)/2]}], {n, 80}]

a(n) = n*sum(i=1, (n-1)\2, (n-i)*(n-2*i)*isprime(i)*isprime(n-i)); \\ Michel Marcus, Apr 21 2018

a(n) = n * Sum_{i=1..floor((n-1)/2)} (n-i) * (n-2*i) * c(i) * c(n-i), where c is the prime characteristic (A010051).

A347823 Triangle read by rows: T(n,k) = (n+k+1)*binomial(n,k), 0 <= k <= n.

Original entry on oeis.org

1, 2, 3, 3, 8, 5, 4, 15, 18, 7, 5, 24, 42, 32, 9, 6, 35, 80, 90, 50, 11, 7, 48, 135, 200, 165, 72, 13, 8, 63, 210, 385, 420, 273, 98, 15, 9, 80, 308, 672, 910, 784, 420, 128, 17, 10, 99, 432, 1092, 1764, 1890, 1344, 612, 162, 19, 11, 120, 585, 1680, 3150, 4032, 3570, 2160, 855, 200, 21

Offset: 0

Jules Beauchamp, Jan 23 2022

			Triangle begins:
  1;
  2,  3;
  3,  8,   5;
  4, 15,  18,   7;
  5, 24,  42,  32,   9;
  6, 35,  80,  90,  50,  11;
  7, 48, 135, 200, 165,  72, 13;
  8, 63, 210, 385, 420, 273, 98, 15;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums give A053220.
Columns give A000027, A005563, A212343.
Diagonals give A005408, A001105, A059270, A112742.
Cf. A007318, A094727.

T(n,k) = (n+k+1)*binomial(n,k) \\ Andrew Howroyd, Jan 23 2022

T(n,k) = A094727(n+1,k)*A007318(n,k).

Row g.f.: (1 + x)^(n-1)*(1 + n + x + 2*n*x). - Stefano Spezia, Jan 23 2022

A137448 Triangle T(n,k) = (1-k(k-1))A053120(n,k), read by rows, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, -1, 0, -2, 0, -3, 0, -20, 1, 0, 8, 0, -88, 0, 5, 0, 100, 0, -304, -1, 0, -18, 0, 528, 0, -928, 0, -7, 0, -280, 0, 2128, 0, -2624, 1, 0, 32, 0, -1760, 0, 7424, 0, -7040, 0, 9, 0, 600, 0, -8208, 0, 23616, 0, -18176, -1, 0, -50, 0, 4400, 0, -32480, 0, 70400, 0, -45568

Offset: 0

Roger L. Bagula and Gary W. Adamson, Apr 18 2008

The entries may also be defined as the coefficient [x^k] of the polynomial T(n,x)-x^2* (d^2/dx^2) T(n,x) where T are the Chebyshev polynomials (A053120).

Row sums are 1, 1, -3, -23, -79, -199, -419, -783, -1343, -2159, -3299, ...

			1;
0, 1;
-1, 0, -2;
0, -3, 0, -20;
1, 0, 8, 0, -88;
0, 5, 0, 100, 0, -304;
-1, 0, -18, 0, 528, 0, -928;
0, -7, 0, -280, 0, 2128, 0, -2624;
1, 0, 32, 0, -1760, 0, 7424, 0, -7040;
0, 9, 0, 600, 0, -8208,0, 23616, 0, -18176;
-1, 0, -50, 0, 4400, 0, -32480, 0, 70400, 0, -45568;

Cf. A053120, A112742.

Clear[p, x, a] p[x, 0] = 1; p[x, 1] = x + 1; p[x_, n_] := p[x, n] = -x^2*D[ChebyshevT[n, x], {x, 2}] + ChebyshevT[n, x]; Table[Expand[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a]

Row sums: Sum_{k=0..n} T(n,k) = (3+n^2-n^4)/3 = 1-A112742(n). - R. J. Mathar, Sep 10 2013

T(1,0) corrected by R. J. Mathar, Sep 10 2013

A373504 Triangular array: row n gives the coefficients T(n,k) of powers x^(2k) in the series expansion of ((b^n + b^(-n))/2)^2, where b = x + sqrt(x^2 + 1).

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 9, 24, 16, 1, 16, 80, 128, 64, 1, 25, 200, 560, 640, 256, 1, 36, 420, 1792, 3456, 3072, 1024, 1, 49, 784, 4704, 13440, 19712, 14336, 4096, 1, 64, 1344, 10752, 42240, 90112, 106496, 65536, 16384, 1, 81, 2160, 22176, 114048, 329472, 559104, 552960, 294912, 65536

Offset: 0

Clark Kimberling, Aug 03 2024

Related to Chebyshev polynomials of the first kind; see A123588.

			First 8 rows:
  1
  1    1
  1    4     4
  1    9    24     16
  1   16    80    128     64
  1   25   200    560    640    256
  1   36   420   1792   3456   3072   1024
  1   49   784   4704  13440  19612  14336  4096
The 4th polynomial is 1 + 9 x^2 + 24 x^4 + 16 x^6.

Cf. A000012 (col 0), A000290 (col 1), A002415 ((1/4)*col(2)), A112742 (col 2), A000302 (T(n,n)), A123588, A008310.
Row sums give A055997(n+1).
Triangle without column 0 gives A334009.

p:= proc(n) option remember; (b-> series(
      ((b^n+b^(-n))/2)^2, x, 2*n+1))(x+sqrt(x^2+1))
    end:
T:= (n, k)-> coeff(p(n), x, 2*k):
seq(seq(T(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Aug 03 2024

t[n_] := ((x + Sqrt[x^2 + 1])^n + (x + Sqrt[x^2 + 1])^(-n))/2
u = Expand[Table[FullSimplify[Expand[t[n]]], {n, 0, 10}]^2]
v = Column[CoefficientList[u, x^2]] (* array *)
Flatten[v] (* sequence *)
T[n_, k_] := If[k==0, 1, 4^(k - 1)*(2*Binomial[n + k, 2*k] - Binomial[n + k -1, 2*k -1])]; Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* Detlef Meya, Aug 11 2024 *)

T(n, k) = if (k=0) then 1, otherwise 4^(k - 1)*(2*binomial(n + k, 2*k) - binomial(n + k - 1, 2*k - 1)). - Detlef Meya, Aug 11 2024

cp's OEIS Frontend

A144216 C(m,2)+C(n,2), m>=1, n>=1: a rectangular array R read by antidiagonals.

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A210220 T(n, k) = -binomial(2*n-k+2, k+1)*hypergeom([2*n-k+3, 1], [k+2], 2). Triangle read by rows, T(n, k) for 1 <= k <= n.

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A350050 a(n) = (2*n^4 - 6*(-1)^n*n^2 - 2*n^2 + 3*(-1)^n - 3)/96.

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A208532 Mirror image of triangle in A125185; unsigned version of A120058.

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A233295 Riordan array ((1+x)/(1-x)^3, 2*x/(1-x)).

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A303231 Total volume of all rectangular prisms with dimensions q, p+q and |q-p| such that p and q are prime, n = p+q and p < q.

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A347823 Triangle read by rows: T(n,k) = (n+k+1)*binomial(n,k), 0 <= k <= n.

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A137448 Triangle T(n,k) = (1-k*(k-1))*A053120(n,k), read by rows, 0<=k<=n.

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A210220 T(n, k) = -binomial(2n-k+2, k+1)hypergeom([2*n-k+3, 1], [k+2], 2). Triangle read by rows, T(n, k) for 1 <= k <= n.

A350050 a(n) = (2n^4 - 6(-1)^nn^2 - 2n^2 + 3*(-1)^n - 3)/96.

A137448 Triangle T(n,k) = (1-k(k-1))A053120(n,k), read by rows, 0<=k<=n.