A371637 -id:A371637 - cp's OEIS frontend

A371683 a(n) = Sum_{k=0..n} (-2)^(3k)binomial(2n, 2k)Euler(2k, 1/2). Row sums of A371637.

1, 3, 33, 819, 37281, 2720643, 291107457, 42945429747, 8354465297601, 2072193715976067, 638269648981638753, 239021193599722872627, 106946291677392350660961, 56346809266835212819000323, 34528790475992735166895973313, 24349545528533035663737512791539

Offset: 0

Peter Luschny, Apr 03 2024

nonn

Cf. A371637, A371684.

seq(add((-8)^k*binomial(2*n, 2*k)*euler(2*k, 1/2), k = 0..n), n = 0..15);

Table[Sum[(-2)^(3*k)*Binomial[2*n,2*k]*EulerE[2*k,1/2],{k,0,n}],{n,0,15}] (* James C. McMahon, Apr 05 2024 *)

a(n) ~ cosh(Pi/(2*sqrt(2))) * 2^(5*n+3) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Apr 03 2024

A371684 a(n) = Sum_{k=0..n} 2^(3k)binomial(2n, 2k)Euler(2k, 1/2). Alternating row sums of A371637.

Original entry on oeis.org

1, -1, 9, -217, 9841, -717841, 76804665, -11330490025, 2204195526241, -546715992537505, 168397490614671849, -63062013420332052985, 28216110792407667898321, -14866226664969958126495921, 9109882748673411939937074969, -6424247756451800785395922510537

Offset: 0

Peter Luschny, Apr 03 2024

sign

Cf. A371637, A371683.

seq(add(2^(3*k)*binomial(2*n, 2*k)*euler(2*k, 1/2), k = 0..n), n = 0..15);

Table[Sum[2^(3*k)*Binomial[2*n, 2*k]*EulerE[2*k, 1/2], {k, 0, n}], {n, 0, 20}] (* Paolo Xausa, Apr 17 2024 *)

a(n) ~ (-1)^n * cos(Pi/(2*sqrt(2))) * 2^(5*n+3) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Apr 03 2024

A372001 Array read by descending antidiagonals: A family of generalized Catalan numbers generated by a generalization of Deléham's Delta operator.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 14, 15, 5, 1, 1, 42, 105, 61, 9, 1, 1, 132, 945, 1385, 297, 17, 1, 1, 429, 10395, 50521, 24273, 1585, 33, 1, 1, 1430, 135135, 2702765, 3976209, 485729, 8865, 65, 1, 1, 4862, 2027025, 199360981, 1145032281, 372281761, 10401345, 50881, 129, 1, 1

Offset: 1

Peter Luschny, Apr 21 2024

Deléham's Delta operator is defined in A084938. It maps two sequences (a, b) to a triangle T. The given sequences are the coefficients of the linear function p = a + x*b which is the starting point of a recurrence described in A084938 and implemented in A371637. The generalization given here extends the number of input sequences to any number, mapping (a, b, c, ...) to p = a + x*b + x^2*c ... but leaves the recurrence unchanged.
The result, as said, is a triangle that we can evaluate in two ways: Firstly, we only return the main diagonal. In this case, we created a new sequence from n given sequences. This case is implemented by the function A(n, dim) below.
Alternatively, we return the entire triangle. But since the triangle is irregular, we convert it into a regular one by taking only every n-th term of a row. This case is handled by the function T(n, dim). For the first few triangles generated this way, see the link section.

			Array starts:
  [0] 1, 1,  2,     5,        14,            42,                132, ...
  [1] 1, 1,  3,    15,       105,           945,              10395, ...
  [2] 1, 1,  5,    61,      1385,         50521,            2702765, ...
  [3] 1, 1,  9,   297,     24273,       3976209,         1145032281, ...
  [4] 1, 1, 17,  1585,    485729,     372281761,       601378506737, ...
  [5] 1, 1, 33,  8865,  10401345,   38103228225,    352780110115425, ...
  [6] 1, 1, 65, 50881, 231455105, 4104215813761, 220579355255364545, ...
.
Seen as a triangle T(n, k) = A(k, n - k):
  [0] [  1]
  [1] [  1,     1]
  [2] [  2,     1,     1]
  [3] [  5,     3,     1,     1]
  [4] [ 14,    15,     5,     1,    1]
  [5] [ 42,   105,    61,     9,    1,  1]
  [6] [132,   945,  1385,   297,   17,  1, 1]
  [7] [429, 10395, 50521, 24273, 1585, 33, 1, 1]

Peter Luschny, First few triangles generated by A372001.

By ascending antidiagonals: A290569.
Family: A000108 (n=0), A001147 (n=1), A000364 (n=2), A216966 (n=3), A227887 (n=4), A337807 (n=5), A337808 (n=6), A337809 (n=7).
Cf. A291333 (main diagonal), A371999 (row sums of triangle).
Cf. A084938, A131427, A371994, A371637.

def GeneralizedDelehamDelta(F, dim, seq=True):  # The algorithm.
    ring = PolynomialRing(ZZ, 'x')
    x = ring.gen()
    A = [sum(F[j](k) * x^j for j in range(len(F))) for k in range(dim)]
    C = [ring(0)] + [ring(1) for i in range(dim)]
    for k in range(dim):
        for n in range(k, 0, -1):
            C[n] = C[n-1] + C[n+1] * A[n-1]
        yield list(C[1])[-1] if seq else list(C[1])
def F(n):  # Define the input functions.
    def p0(): return lambda n: pow(n, n^0)
    def p(k): return lambda n: pow(n + 1, k)
    return [p0()] + [p(k) for k in range(n + 1)]
def A(n, dim): # Return only the main diagonal of the triangle.
    return [r for r in GeneralizedDelehamDelta(F(n), dim)]
for n in range(7): print(A(n, 7))
def T(n, dim): # Return the regularized triangle.
    R = GeneralizedDelehamDelta(F(n), dim, False)
    return [[r[k] for k in range(0, len(r), n + 1)] for r in R]
for n in range(0, 4):
    for row in T(n, 6): print(row)

A = DELTA([x -> (x + 1)^k : 0 <= k <= n]), i.e. here the input functions of the generalized Delta operator are the (shifted) power functions. The returned sequence is the main diagonal of the generated triangle.

A371994 Triangle read by rows: Related to the Euler numbers.

Original entry on oeis.org

1, 0, 1, 0, 1, 5, 0, 1, 19, 61, 0, 1, 42, 498, 1385, 0, 1, 74, 1932, 19238, 50521, 0, 1, 115, 5290, 114830, 1057475, 2702765, 0, 1, 165, 11805, 449539, 8949633, 79160457, 199360981, 0, 1, 224, 23016, 1360198, 47306246, 899141244, 7768928932, 19391512145

Offset: 0

Peter Luschny, Apr 21 2024

Inspired by Philippe Deléham's formula for A371637.

			Triangle starts:
  [0] [1]
  [1] [0, 1]
  [2] [0, 1,   5]
  [3] [0, 1,  19,    61]
  [4] [0, 1,  42,   498,   1385]
  [5] [0, 1,  74,  1932,  19238,   50521]
  [6] [0, 1, 115,  5290, 114830, 1057475,  2702765]
  [7] [0, 1, 165, 11805, 449539, 8949633, 79160457, 199360981]

Cf. A000364 (main diagonal), A371637, A371765 (row sums), A372001.

Cf. A084938.

# Using function GeneralizedDelehamDelta from A372001.
def A371994_triangle(dim):
    a = lambda n: n % 2
    b = lambda n: (n + 1)^2
    return GeneralizedDelehamDelta([a, b], dim, False)
for row in A371994_triangle(8): print(row)

Triangle T given by [0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 4, 9, 16, 25, 36, ...] where DELTA is the operator defined by Deléham in A084938.

cp's OEIS Frontend

A371683 a(n) = Sum_{k=0..n} (-2)^(3k)binomial(2n, 2k)Euler(2k, 1/2). Row sums of A371637.

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A371684 a(n) = Sum_{k=0..n} 2^(3k)binomial(2n, 2k)Euler(2k, 1/2). Alternating row sums of A371637.

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A372001 Array read by descending antidiagonals: A family of generalized Catalan numbers generated by a generalization of Deléham's Delta operator.

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SageMath

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A371994 Triangle read by rows: Related to the Euler numbers.

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cp's OEIS Frontend

A371683 a(n) = Sum_{k=0..n} (-2)^(3*k)*binomial(2*n, 2*k)*Euler(2*k, 1/2). Row sums of A371637.

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Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A371684 a(n) = Sum_{k=0..n} 2^(3*k)*binomial(2*n, 2*k)*Euler(2*k, 1/2). Alternating row sums of A371637.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A372001 Array read by descending antidiagonals: A family of generalized Catalan numbers generated by a generalization of Deléham's Delta operator.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

SageMath

Formula

A371994 Triangle read by rows: Related to the Euler numbers.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

SageMath

Formula

A371683 a(n) = Sum_{k=0..n} (-2)^(3k)binomial(2n, 2k)Euler(2k, 1/2). Row sums of A371637.

A371684 a(n) = Sum_{k=0..n} 2^(3k)binomial(2n, 2k)Euler(2k, 1/2). Alternating row sums of A371637.