A194589 -id:A194589 - cp's OEIS frontend

A194588 a(n) = A189912(n-1)-a(n-1) for n>0, a(0) = 1; extended Riordan numbers.

1, 0, 2, 2, 8, 17, 49, 128, 356, 983, 2759, 7779, 22087, 63000, 180478, 518846, 1496236, 4326383, 12539335, 36419069, 105971473, 308866226, 901573732, 2635235789, 7712078755, 22594899002, 66266698424, 194531585078, 571561286576, 1680679630089, 4945738222801

Offset: 0

Peter Luschny, Aug 30 2011

nonn

Peter Luschny, The lost Catalan numbers.

Cf. A005043, A194589.

A189912 := n -> add(n!/((n-k)!*iquo(k,2)!^2 *(iquo(k,2)+1)),k=0..n):
A194588 := n -> `if`(n=0,1,A189912(n-1)-A194588(n-1)):

a[0] = 1; a[n_] := a[n] = Sum[(n-1)!/((n-k-1)!*Quotient[k, 2]!^2*(1 + Quotient[k, 2])), {k, 0, n-1}] - a[n-1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 30 2013 *)

a(n) = ((n+1) mod 2) + (1/2)*sum_{k=1..n}((-1)^k*binomial(n,k)*((k+1)/2)^(k mod 2)*(k+1)$+2*(-1)^n*(2*k)$/(k+1)), where n$ denotes the swinging factorial A056040(n).

A194590 a(n) = (-1)^n(A056040(n+1)A152271(n)-2^n)/2.

Original entry on oeis.org

0, 0, 1, -2, 7, -14, 38, -76, 187, -374, 874, -1748, 3958, -7916, 17548, -35096, 76627, -153254, 330818, -661636, 1415650, -2831300, 6015316, -12030632, 25413342, -50826684, 106853668, -213707336, 447472972, -894945944, 1867450648, -3734901296, 7770342787

Offset: 0

Peter Luschny, Aug 30 2011

sign

The binomial transform of a(n) are the complementary Riordan numbers A194589 (see link).

Peter Luschny, The lost Catalan numbers

Cf. A107373 (has offset 1).

A056040 := n -> n!/iquo(n,2)!^2:
A152271 := n -> `if`(n mod 2 = 0, 1, (n+1)/2):
A194590 := n -> (-1)^n*(A056040(n+1)*A152271(n)-2^n)/2:

sf[n_] := n!/Quotient[n, 2]!^2;
a[n_] := (-1)^n (sf[n + 1] * If[EvenQ[n], 1, (n + 1)/2] - 2^n)/2;
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Jun 26 2019 *)

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*cr(k), where cr(k) are the complementary Riordan numbers A194589.

cp's OEIS Frontend

A194588 a(n) = A189912(n-1)-a(n-1) for n>0, a(0) = 1; extended Riordan numbers.

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A194590 a(n) = (-1)^n(A056040(n+1)A152271(n)-2^n)/2.

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cp's OEIS Frontend

A194588 a(n) = A189912(n-1)-a(n-1) for n>0, a(0) = 1; extended Riordan numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A194590 a(n) = (-1)^n*(A056040(n+1)*A152271(n)-2^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A194590 a(n) = (-1)^n(A056040(n+1)A152271(n)-2^n)/2.