A059738 -id:A059738 - cp's OEIS frontend

A375181 Inverse binomial transform of A376277.

1, 1, 2, 3, 6, 12, 28, 70, 189, 533, 1551, 4599, 13807, 41784, 127144, 388373, 1189661, 3651910, 11228851, 34571914, 106555917, 328713138, 1014797705, 3134841053, 9689148780, 29961083746, 92683964271, 286816872102, 887849075464, 2749110140301, 8514323952447

Offset: 0

Thomas Scheuerle, Sep 23 2024

The Hankel sequence transform gives {1, 1, 1, 1, 1, ...}.
The Hankel sequence transform starting with the second term gives {1, -3, -2, 1, 3, 2, -1, -3, -2, ...}.
The periodic pattern in the continued fractions of the generating function starts after some prefix, this causes high complexity in the generating function and makes a nice combinatorial interpretation less likely, therefore the keyword "less" was considered.

Cf. A059738, A376277.

my(N=30, x='x+O('x^N)); Vec((2*x^2-sqrt(-3*x^2-2*x+1)+3*x-1)/(-5*x^3+sqrt(-3*x^2-2*x+1)*(x^2+x-1)+4*x-1))

G.f.: 1/(1-1*x/(1-1*x/(1+1*x/(1+1*x/(1-3*x/(1-(1/3)*x/(1-(2/3)*x/(1-(3/2)*x/(1+(1/2)*x/(...)))))))))), a continued fraction expansion. The coefficients of x are {-1, -1, 1, 1, -3, -(1/3), -(2/3), -(3/2), (1/2), 2, -3, ...}. The numerators will repeat {1, 2, 3} the denominators {1, 1, 2, 2, 3, 3} the sign repeats {-,-,-,-,+,+}.
G.f.: (2*x^2 - sqrt(-3*x^2 - 2*x + 1) + 3*x - 1)/(-5*x^3 + sqrt(-3*x^2 - 2*x + 1)*(x^2 + x - 1) + 4*x - 1)
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*A376277(k).
D-finite with recurrence (-n+3)*a(n) +(7*n-25)*a(n-1) +2*(-6*n+25)*a(n-2) +10*(-n+4)*a(n-3) +2*(16*n-73)*a(n-4) +(n-5)*a(n-5) +21*(-n+5)*a(n-6)=0. - R. J. Mathar, Oct 24 2024

A171616 Triangle T : T(n,k)= binomial(n,k)*A000957(n+1-k).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 6, 8, 6, 0, 1, 18, 30, 20, 10, 0, 1, 57, 108, 90, 40, 15, 0, 1, 186, 399, 378, 210, 70, 21, 0, 1, 622, 1488, 1596, 1008, 420, 112, 28, 0, 1, 2120, 5598, 6696, 4788, 2268, 756, 168, 36, 0, 1, 7338, 21200, 27990, 22320, 11970, 4536, 1260, 240, 45

Offset: 0

Philippe Deléham, Dec 13 2009

			Triangle begins : 1 ; 0,1 ; 1,0,1 ; 2,3,0,1 ; 6,8,6,0,1 ; 18,30,20,10,0,1 ; ...

Cf. A007318

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000957(n+1), A033321(n), A033543(n) for x = 0,1,2 respectively. Sum_{k, 0<=k<=n} T(n,k)*(-1)^(n-k)*x^k = A054341(n), A059738(n), A049027(n+1) for x = 2,3,4 respectively.

A375723 Square array read by antidiagonals, where the top row is the powers of 2 (A000079) and the other numbers are the sum of the neighbors in the preceding row.

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 8, 14, 24, 34, 16, 28, 49, 83, 117, 32, 56, 98, 171, 288, 405, 64, 112, 196, 343, 597, 1002, 1407, 128, 224, 392, 686, 1200, 2085, 3492, 4899, 256, 448, 784, 1372, 2401, 4198, 7285, 12184, 17083, 512, 896, 1568, 2744, 4802, 8403, 14686, 25463, 42546, 59629

Offset: 0

WG Zeist, Aug 25 2024

Each number in the top row of the array is determined by the pre-defined sequence (in this case, the powers of 2, A000079). Each number in lower rows is the sum of the numbers vertically or diagonally above it (so, the number at the left end of each row is the sum of two numbers, and all other numbers the sum of three). This is the same method as for constructing A217536, which has the top row be the nonnegative integers instead; other similar arrays are described in the comments of that sequence.
The main diagonal is the powers of 7 (A000420), and all numbers above or to the right of the main diagonal are multiples of powers of 2 and powers of 7. Specifically, the number in row m and column n, for n >= m, is 2^(n-m) * 7^m. Above the main diagonal, all numbers in the same column have the same final digit in base 10, and all numbers are 7/2 times the number immediately above.
More broadly, for any similarly constructed array with the powers of x as the top row, then the main diagonal will be the powers of (x^2 + x + 1) and the numbers above the main diagonal will be x^(n-m) * (x^2 + x + 1)^m (see also A062105, which can be interpreted as a similar array with the powers of 1 in the top row, and A020474 with the powers of 0).

			The array starts:
  1  2  4   8
  3  7  14  28
  10 24 49  98
  34 83 171 343

The main diagonal gives A000420 (powers of 7). The first column gives A059738.

Cf. A217536, A062105, A020474.

T(m+1,n) = sum(T(m,k), |k-n| <= 1) (and T(0,n)=2^n), m, n >= 0.

cp's OEIS Frontend

A375181 Inverse binomial transform of A376277.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

A171616 Triangle T : T(n,k)= binomial(n,k)*A000957(n+1-k).

Views

Author

Keywords

Examples

Crossrefs

Formula

A375723 Square array read by antidiagonals, where the top row is the powers of 2 (A000079) and the other numbers are the sum of the neighbors in the preceding row.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula