cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A069514 Numbers n such that sigma(reversal(n)) = reversal(sigma(n)). Ignore leading 0's.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 14, 41, 124, 194, 333, 421, 491, 1324, 4231, 13324, 17054, 17571, 42331, 45071, 120530, 138465, 386650, 564831, 1130324, 1216360, 1333324, 1727571, 1757271, 1757571, 1787871, 2249422, 4230311, 4233331, 4369634
Offset: 1

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Author

Joseph L. Pe, Apr 15 2002

Keywords

Comments

For an arithmetical function f, call the arguments n such that f(reverse(n)) = reverse(f(n)) the "palinpoints" of f. This sequence is the sequence of palinpoints of f(n) = sigma(n).
If n is in the sequence and 10 doesn't divide n then the reversal of n is also in the sequence. - Farideh Firoozbakht, Aug 31 2004
Comments from Farideh Firoozbakht, Jan 16 2005. "The largest term that I found is M=(58*100^687 - 157)/33; the length of M is 1375. I proved the following facts about this sequence:
"I : If p=(58*100^n - 157)/99 is prime then 3*p is in the sequence, the sequence A102285 gives such n's.
"II : If p=(59*100^n - 257)/99 is prime then 3*p is in the sequence, I found only two primes of this form the first for n=3 and the second for n=27, next such n is greater than 3400.
"III : If both numbers p=10^n - 3 & q=5*10^n - 9 are primes then both numbers 2*p & q are in the sequence, q is reversal of 2*p. I found only two such n's, n=1 & 2.
"IV : If both numbers p=(10^n-7)/3 & q=(127*10^(n-1)-7)/3 are primes then both numbers 4*p & q are in the sequence, q is the reversal of 4*p, the sequence A102287 are these terms of A069514, I found only four such n's, n=2,3,4 & 6."

Examples

			Let f(n) = sigma(n). Then f(194) = 294, f(491) = 492, so f(reverse(194)) = reverse(f(194)). Therefore 194 belongs to the sequence.

Crossrefs

Cf. A102285, A102286, A102287.

Programs

  • Mathematica
    rev[n_] := FromDigits[Reverse[IntegerDigits[n]]]; f[n_] := DivisorSigma[1, n]; Select[Range[10^6], f[rev[ # ]] == rev[f[ # ]] &]

Extensions

More terms from Farideh Firoozbakht, Aug 31 2004

A224271 Number of set partitions of {1,2,...,n} such that the element 1 is in an odd-sized block.

Original entry on oeis.org

1, 1, 3, 8, 28, 107, 459, 2151, 10931, 59700, 348146, 2155925, 14112377, 97266301, 703484851, 5323515156, 42040470092, 345670438963, 2953171501547, 26166317121747, 240047041176843, 2276607815242880, 22290187889601330, 225018607554567149, 2339331996135377345
Offset: 1

Views

Author

Geoffrey Critzer, Apr 02 2013

Keywords

Examples

			a(4) = 8 because we have: {{1},{2,3,4}}, {{1,3,4},{2}}, {{1,2,3},{4}}, {{1,2,4},{3}}, {{1},{2},{3,4}}, {{1},{2,3},{4}}, {{1},{2,4},{3}}, {{1},{2},{3},{4}}.

Links

Crossrefs

Cf. A000110, A003724, A005046, A102286, A124322, A131719, A283424.

Programs

  • Maple
    with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*`if`(irem(i, 2)=0, x^j, 1), j=0..n/i))))
    end:
a:= n-> (p-> add(coeff(p, x, i)*(i+1), i=0..degree(p)))(b(n-1$2)):
seq(a(n), n=1..15);  # Alois P. Heinz, Mar 08 2015
# second Maple program:
b:= proc(n, t, m) option remember; `if`(n=0, t, (m-1)*
      b(n-1, t, m)+b(n-1, 1-t, m)+b(n-1, t, m+1))
    end:
a:= n-> b(n-1, 1$2):
seq(a(n), n=1..25);  # Alois P. Heinz, May 17 2023
  • Mathematica
    nn=25;Drop[Range[0,nn]!CoefficientList[Series[Integrate[Exp[Cosh[x]-1]D[ Exp[Sinh[x]],x],x],{x,0,nn}],x],1]

Formula

E.g.f. A(x) satisfies: A'(x) = B'(x)*C(x) where B(x) is the e.g.f. for A003724 and C(x) is the e.g.f. for A005046.
a(n) = Sum_{k=0..floor((n-1)/2)} (k+1)*A124322(n-1,k). - Alois P. Heinz, Apr 02 2013
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * Bell(n-2*k-1). - Ilya Gutkovskiy, Apr 10 2022
From Alois P. Heinz, May 17 2023: (Start)
a(n) = Sum_{k=0..n-1} (-1)^k * A283424(n-1,k).
a(n) mod 2 = A131719(n+1). (End)

A124321 Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} (or of any n-set) having k blocks of odd size (0<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 4, 0, 1, 4, 0, 10, 0, 1, 0, 31, 0, 20, 0, 1, 31, 0, 136, 0, 35, 0, 1, 0, 379, 0, 441, 0, 56, 0, 1, 379, 0, 2500, 0, 1176, 0, 84, 0, 1, 0, 6556, 0, 11740, 0, 2730, 0, 120, 0, 1, 6556, 0, 59671, 0, 43870, 0, 5712, 0, 165, 0, 1, 0, 150349, 0, 378356, 0, 138622, 0
Offset: 0

Views

Author

Emeric Deutsch, Oct 28 2006

Keywords

Comments

Row sums are the Bell numbers (A000110).
Sum_{k=0..n} k*T(n,k) = A102286(n).
T(2*n,0) = A005046(n); T(2*n+1,0) = 0.

Examples

			T(3,1) = 4 because we have 123, 1|23, 12|3 and 13|2.
Triangle starts:
  1;
  0,  1;
  1,  0,  1;
  0,  4,  0,  1;
  4,  0, 10,  0,  1;
  0, 31,  0, 20,  0,  1;

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 225.

Links

Crossrefs

Cf. A000110, A102286, A005046, A124322.

Programs

  • Maple
    G:=exp(t*sinh(z)+cosh(z)-1): Gser:=simplify(series(G,z=0,15)): for n from 0 to 12 do P[n]:=sort(n!*coeff(Gser,z,n)) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*`if`(irem(i, 2)=1, x^j, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..15);  # Alois P. Heinz, Mar 08 2015
  • Mathematica
    nn = 10; Range[0, nn]! CoefficientList[Series[Exp[ (Cosh[x] - 1) + y Sinh[x]], {x, 0, nn}], {x, y}] // Grid (* Geoffrey Critzer, Aug 28 2012 *)

Formula

E.g.f.: G(t,z) = exp(t*sinh(z)+cosh(z)-1).

A352905 Expansion of e.g.f. sin(x) * exp(exp(x) - 1).

Original entry on oeis.org

0, 1, 2, 5, 16, 56, 218, 937, 4376, 22027, 118744, 681570, 4144988, 26598313, 179451366, 1268930969, 9378332608, 72267300476, 579336907254, 4822070246225, 41597773001612, 371306237988959, 3424303740576440, 32583334570211654, 319487530199710232, 3224337031346853361
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 07 2022

Keywords

Comments

The first negative term is a(71).

Crossrefs

Cf. A000110, A009542, A102286, A351745.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Sin[x] Exp[Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k Binomial[n, 2 k + 1] BellB[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}], {n, 0, 25}]

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n,2*k+1) * Bell(n-2*k-1).
Conjecture: a(n) = (i/(2*e))*Sum_{k=0..oo} ((k - i)^n - (k + i)^n)/(k!), where i = sqrt(-1) and e = exp(1). - Velin Yanev, Jul 06 2024
Showing 1-4 of 4 results.

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