A010748 -id:A010748 - cp's OEIS frontend

A143983 Triangle T(n,k), n>=1, 1<=k<=n, read by rows, where sequence a_k of column k has a_k(0)=1, followed by (k-1)-fold 0 and a_k(n) shifts k places down under binomial transform.

1, 2, 1, 5, 1, 1, 15, 2, 1, 1, 52, 5, 1, 1, 1, 203, 13, 2, 1, 1, 1, 877, 36, 6, 1, 1, 1, 1, 4140, 109, 17, 2, 1, 1, 1, 1, 21147, 359, 44, 7, 1, 1, 1, 1, 1, 115975, 1266, 112, 23, 2, 1, 1, 1, 1, 1, 678570, 4731, 304, 65, 8, 1, 1, 1, 1, 1, 1, 4213597, 18657, 918, 165, 30, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Alois P. Heinz, Sep 06 2008

The matrix inverse starts:
     1;
    -2,   1;
    -3,  -1,  1;
    -8,  -1, -1,  1;
   -31,  -3,  0, -1,  1;
  -132,  -7, -1,  0, -1,  1;
  -616, -19, -4,  0,  0, -1, 1; - R. J. Mathar, Mar 22 2013

			T(5,2) = 5, because [1,3,3,1] * [1,0,1,1] = 5.
Triangle begins:
:      1;
:      2,    1;
:      5,    1,   1;
:     15,    2,   1,  1;
:     52,    5,   1,  1, 1;
:    203,   13,   2,  1, 1, 1;
:    877,   36,   6,  1, 1, 1, 1;
:   4140,  109,  17,  2, 1, 1, 1, 1;
:  21147,  359,  44,  7, 1, 1, 1, 1, 1;
: 115975, 1266, 112, 23, 2, 1, 1, 1, 1, 1;

Alois P. Heinz, Rows n = 1..141, flattened
N. J. A. Sloane, Transforms

Columns 1-6 give: A000110, A000994, A000996, A010748, A010749, A010750.

Cf. A007318.

T:= proc(n, k) option remember; `if`(n

t[n_, k_] := t[n, k] = If[n < k, If[n == 0, 1, 0], Sum[Binomial[n-k, j]*t[j, k], {j, 0, n-k}]]; Table[Table[t[n, k], {k, 1, n}], {n, 1, 13}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)

T(n,k) = Sum_{j=0..n-k} C(n-k,j)*T(j,k) if n>=k, else T(n,k) = 1 if n=1, else T(n,k) = 0.

A351343 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 * A(x/(1 - 2x)) / (1 - 2x).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 9, 27, 81, 245, 761, 2493, 8849, 34519, 147057, 670327, 3198561, 15732905, 79174929, 407127897, 2145061729, 11635963499, 65309080185, 380583443187, 2304629301041, 14475031232285, 93943897651017, 627220447621973, 4290783719133041, 29988917377046207

Offset: 0

Ilya Gutkovskiy, Feb 08 2022

nonn

Shifts 4 places left under 2nd-order binomial transform.

Cf. A004211, A007472, A010748, A210541, A275934, A351342, A351344, A351345.

nmax = 29; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 4, 1, Sum[Binomial[n - 4, k] 2^k a[n - k - 4], {k, 0, n - 4}]]; Table[a[n], {n, 0, 29}]

a(0) = ... = a(3) = 1; a(n) = Sum_{k=0..n-4} binomial(n-4,k) * 2^k * a(n-k-4).

cp's OEIS Frontend

A143983 Triangle T(n,k), n>=1, 1<=k<=n, read by rows, where sequence a_k of column k has a_k(0)=1, followed by (k-1)-fold 0 and a_k(n) shifts k places down under binomial transform.

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A351343 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 * A(x/(1 - 2x)) / (1 - 2x).

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cp's OEIS Frontend

A143983 Triangle T(n,k), n>=1, 1<=k<=n, read by rows, where sequence a_k of column k has a_k(0)=1, followed by (k-1)-fold 0 and a_k(n) shifts k places down under binomial transform.

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Maple

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Formula

A351343 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 * A(x/(1 - 2*x)) / (1 - 2*x).

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A351343 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 * A(x/(1 - 2x)) / (1 - 2x).