A342362 -id:A342362 - cp's OEIS frontend

A342354 M(n,k) = 2n^2 + 2k + 1 for 0 <= k <= n and M(n,k) = 2k^2 + 4k - 2*n + 1 for 0 <= n <= k; square array M(n,k) read by ascending antidiagonals (n, k >= 0).

1, 3, 7, 9, 5, 17, 19, 11, 15, 31, 33, 21, 13, 29, 49, 51, 35, 23, 27, 47, 71, 73, 53, 37, 25, 45, 69, 97, 99, 75, 55, 39, 43, 67, 95, 127, 129, 101, 77, 57, 41, 65, 93, 125, 161, 163, 131, 103, 79, 59, 63, 91, 123, 159, 199, 201, 165, 133, 105, 81, 61, 89, 121, 157, 197, 241, 243, 203, 167, 135, 107, 83, 87, 119, 155, 195, 239, 287

Offset: 0

Petros Hadjicostas, Mar 08 2021

This is a square array defined by J. M. Bergot in A005917 (originally by mistake in A047926). Here is the edited description of the array by this contributor.

Construct an array M with M(0,n) = 2*n^2 + 4*n + 1 = A056220(n+1), M(n,0) = 2*n^2 + 1 = A058331(n) and M(n,n) = 2*n*(n+1) + 1 = A001844(n). Row(n) begins with all the increasing odd numbers from A058331(n) to A001844(n) and column(n) begins with all the decreasing odd numbers from A056220(n+1) to A001844(n). The sum of the terms in row(n) plus those in column(n) minus M(n,n) equals A005917(n+1).

			Square array M(n,k) (n, k >= 0) begins:
   1,  7, 17, 31, 49, 71, 97, 127, ...
   3,  5, 15, 29, 47, 69, 95, 125, ...
   9, 11, 13, 27, 45, 67, 93, 123, ...
  19, 21, 23, 25, 43, 65, 91, 121, ...
  33, 35, 37, 39, 41, 63, 89, 119, ...
  51, 53, 55, 57, 59, 61, 87, 117, ...
  73, 75, 77, 79, 81, 83, 85, 115, ...
  ...
The triangular array T(n,k) = M(n-k,k) (with rows n >= 0 and columns k = 0..n) is obtained by reading array M by ascending antidiagonals:
   1;
   3,  7;
   9,  5, 17;
  19, 11, 15, 31;
  33, 21, 13, 29, 49;
  51, 35, 23, 27, 47, 71;
  73, 53, 37, 25, 45, 69, 97;
  99, 75, 55, 39, 43, 67, 95, 127;
  ...

Cf. A001844, A005917, A047926, A056220, A058331.

Antidiagonal sums are in A342362.

tabl(nn) = {my(M=matrix(nn+1,nn+1)); for(n=1, nn+1, for(k=1, nn+1, M[n,k]=if(k == n, 2*n^2-2*n+1, if(k < n, 2*n^2-4*n+2*k+1, 2*k^2-2*n+1)))); M}

O.g.f. for rectangular M: (x^4*y^4 + 4*x^3*y^4 + 3*x^4*y^2 - 18*x^3*y^3 - x^2*y^4 + 8*x^3*y^2 + 4*x^2*y^3 - 10*x^3*y + 10*x^2*y^2 - 2*x*y^3 + 8*x^2*y + 4*x*y^2 + 3*x^2 - 18*x*y - y^2 + 4*y + 1)/((1 - x)^3*(1 - y)^3*(1 - x*y)^2).

O.g.f. for triangular T: (x^8*y^4 + 4*x^7*y^4 - x^6*y^4 - 18*x^6*y^3 + 3*x^6*y^2 + 4*x^5*y^3 + 8*x^5*y^2 - 2*x^4*y^3 + 10*x^4*y^2 - 10*x^4*y + 4*x^3*y^2 + 8*x^3*y - x^2*y^2 - 18*x^2*y + 3*x^2 + 4*x*y + 1)/((1 - x)^3*(1 - x*y)^3*(1 - x^2*y)^2).

A342397 Expansion of the o.g.f. (2x^2 + 3x + 2)x/((x + 1)^2(x - 1)^4).

Original entry on oeis.org

0, 2, 7, 18, 35, 62, 98, 148, 210, 290, 385, 502, 637, 798, 980, 1192, 1428, 1698, 1995, 2330, 2695, 3102, 3542, 4028, 4550, 5122, 5733, 6398, 7105, 7870, 8680, 9552, 10472, 11458, 12495, 13602, 14763, 15998, 17290, 18660, 20090, 21602, 23177, 24838, 26565, 28382, 30268, 32248, 34300, 36450

Offset: 0

Petros Hadjicostas, Mar 10 2021

nonn

One-half of the antidiagonal sums of array A220508.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A004126, A169607, A220508, A342462.

CoefficientList[Series[(2x^2+3x+2) x/((x+1)^2 (x-1)^4),{x,0,70}],x] (* or *) LinearRecurrence[{2,1,-4,1,2,-1},{0,2,7,18,35,62},70] (* Harvey P. Dale, Jul 08 2023 *)

/* First program */
seq1(n)={my(x='x+O('x^n)); Vec((2*x^2 + 3*x + 2)*x/((x + 1)^2*(x - 1)^4), -n)}
/* Second program (array M is A220508) */
seq2(nn) = {my(M=matrix(nn+1, nn+1)); my(a=vector(nn+1)); for(n=1, nn+1, for(k=1, nn+1, M[n, k]=if(k == n, n^2-n, if(k < n, n^2-2*n+k, k^2-n)))); for(n=1, nn+1, a[n] = sum(k=1, n, M[n-k+1,k])/2); a}

a(n) = (n+1)*(1 - (-1)^n)/16 + (7/4)*(binomial(n+3, 3) - binomial(n+2, 2)).
a(n) = (A342362(n) - (n + 1))/4.
a(2*n) = A169607(n) and a(2*n + 1) = 2*A004126(n + 1).
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n > 5. - Chai Wah Wu, Mar 11 2021

cp's OEIS Frontend

A342354 M(n,k) = 2n^2 + 2k + 1 for 0 <= k <= n and M(n,k) = 2k^2 + 4k - 2*n + 1 for 0 <= n <= k; square array M(n,k) read by ascending antidiagonals (n, k >= 0).

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A342397 Expansion of the o.g.f. (2x^2 + 3x + 2)x/((x + 1)^2(x - 1)^4).

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cp's OEIS Frontend

A342354 M(n,k) = 2*n^2 + 2*k + 1 for 0 <= k <= n and M(n,k) = 2*k^2 + 4*k - 2*n + 1 for 0 <= n <= k; square array M(n,k) read by ascending antidiagonals (n, k >= 0).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A342397 Expansion of the o.g.f. (2*x^2 + 3*x + 2)*x/((x + 1)^2*(x - 1)^4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A342354 M(n,k) = 2n^2 + 2k + 1 for 0 <= k <= n and M(n,k) = 2k^2 + 4k - 2*n + 1 for 0 <= n <= k; square array M(n,k) read by ascending antidiagonals (n, k >= 0).

A342397 Expansion of the o.g.f. (2x^2 + 3x + 2)x/((x + 1)^2(x - 1)^4).