A277355 -id:A277355 - cp's OEIS frontend

A375042 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = n^2 x and t(x) = 2x+1. See Comments.

1, 2, 1, 5, 6, 1, 11, 36, 36, 1, 22, 157, 432, 396, 1, 40, 553, 3258, 8172, 7128, 1, 67, 1633, 18189, 96138, 227772, 192456, 1, 105, 4179, 80243, 787320, 3881016, 8847792, 7313328, 1, 156, 9534, 293372, 4879713, 44034336, 206779608, 458550720, 372979728, 1

Offset: 1

Clark Kimberling, Sep 11 2024

See A374848 for the definition of obverse convolution and a guide to related sequences and arrays.

			First 3 polynomials in s(x)**t(x) are
  1 + 2x,
  1 + 5 x + 6 x^2,
  1 + 11 x + 36 x^2 + 36 x^3.
First 5 rows of array:
  1   2
  1   5      6
  1   11    36    36
  1   22   157   432   396
  1   40   553  3258  8172  7128

Cf. A000290, A277355 ((1/2)T(n,n+1)), A374848, A375041, A375043.

s[n_] := n^2  x; t[n_] := 1 + 2 x;
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[Expand[u[n]], {n, 0, 10}]
Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]    (* sequence *)

A333938 Decimal expansion of Product_{k>=1} (1 - k/2^k).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Apr 11 2020

			0.0788983500212491810064185920122947774736728143458491250873967214687...

Cf. A048651, A065446, A277355.

evalf(Product(1 - k/2^k, k = 1..infinity), 100);

Join[{0},RealDigits[Product[1-n/2^n,{n,500}],10,120][[1]]] (* Harvey P. Dale, Jan 12 2024 *)

PARI
```
prodinf(k=1, 1 - k/2^k)
```

Equals exp(-Sum_{j>=1} polylog(-j, 1/2^j)/j).

cp's OEIS Frontend

A375042 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = n^2 x and t(x) = 2x+1. See Comments.

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