Revision History for A356900
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#11 by Peter Luschny at Sat Sep 03 08:13:12 EDT 2022
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#10 by Peter Luschny at Sat Sep 03 07:59:22 EDT 2022
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| COMMENTS
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Other special values of this Euler type polynomials are: P(n, -1) = A000364(n); P(n, -1/2) = A002105(n); P(n, 1) = A094088(n). ), where we always make the assumption that the offset of the sequences is 0. A partition refinement of Joffe's triangle A241171 is A327022.
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proposed
editing
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Discussion
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| Sat Sep 03
| 08:12
| Peter Luschny: Note to myself: " ...where we always make the assumption that the offset of the sequences is 0", which unfortunately is not always the case. For example, it would have been much better to define A002105 as Woan did, instead of using this crumpled Bernoulli formula: "The sequence 1, 0, 1, 0, 4, 0, 34, 0, 496, ... counts increasing complete binary trees with e.g.f. sec^2(x/sqrt 2)."
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#9 by Peter Luschny at Sat Sep 03 06:37:40 EDT 2022
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#8 by Peter Luschny at Sat Sep 03 06:37:36 EDT 2022
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| COMMENTS
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Other special values of this Euler type polynomials are: P(n, -1) = A000364(n); P(n, -1/2) = A002105(n); P(n, 1/2) = this sequence; P(n, 1) = A094088(n). A partition refinement of Joffe's triangle A241171 is A327022.
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proposed
editing
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#7 by Peter Luschny at Sat Sep 03 06:32:06 EDT 2022
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#6 by Peter Luschny at Sat Sep 03 06:26:52 EDT 2022
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| COMMENTS
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Other special values of this Euler type polynomials are: P(n, -1) = A000364(n); x P(n, -1/2) = A002105(n); P(n, 1/2) = this sequence; P(n, 1) = A094088(n). A partition refinement of Joffe's triangle A241171 is A327022.
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| MAPLE
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a := n -> 2^n*add((-1)^k*(A241171(n, k)*(-)*(1/2)^k, k = 0..n):
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#5 by Peter Luschny at Sat Sep 03 06:20:58 EDT 2022
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| NAME
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a(n) = P(n, 1/2^) where P(n, x) = x^(-n*)*Sum_{k=0..n} (-1)^k*} A241171(n, k)*(-1/2)^)*x^k.
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| COMMENTS
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Other special values of this Euler type polynomials are: P(n, -1) = A000364(n); x P(n, -1/2) = A002105(n); P(n, 1/2) = this sequence; P(n, 1) = A094088(n). A partition refinement of Joffe's triangle A241171 is A327022.
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| CROSSREFS
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Cf. A241171, A327022, A269941A000364, A002105, A094088, A269941.
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#4 by Peter Luschny at Sat Sep 03 05:34:55 EDT 2022
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euler-joffe, complement to reduced tangent numbers
a(n) = 2^n*Sum_{k=0..n} (-1)^k*A241171(n, k)*(-1/2)^k.
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| MAPLE
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a := n -> 2^n*add((-1)^k*A241171(n, k)*(-1/2)^k, k = 0..n):
seq(a(n), n = 0..16);
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| CROSSREFS
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Cf. A241171, A269941, A002105.
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#3 by Peter Luschny at Sat Sep 03 04:40:25 EDT 2022
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allocatedeuler-joffe, complement to forreduced Petertangent Luschnynumbers
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| DATA
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1, 1, 8, 154, 5552, 321616, 27325088, 3200979664, 494474723072, 97390246272256, 23820397371219968, 7083386168647642624, 2516691244849530785792, 1052914814802404260765696, 512347915163742179541659648, 286902390859642414913802102784, 183187476890368376930869730803712
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| OFFSET
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0,3
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| PROG
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(SageMath) # Using function PtransMatrix from A269941.
def E(n, v):
eulr = lambda n: 1 / ((2 * n - 1) * (2 * n))
norm = lambda n, k: (1 / v)^n * factorial(2 * n)
P = PtransMatrix(n, eulr, norm)
return [(-1)^j * sum([v^k * P[j][k] for k in range(j + 1)]) for j in range(n)]
A356900List = lambda n: E(n, -1/2); print(A356900List(17))
# A002105List = lambda n: E(n, 1/2) returns the reduced tangent numbers A002105.
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| CROSSREFS
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Cf. A269941, A002105.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Peter Luschny, Sep 03 2022
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| STATUS
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approved
editing
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#2 by Peter Luschny at Sat Sep 03 04:26:58 EDT 2022
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| KEYWORD
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allocating
allocated
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