A037952 -id:A037952 - cp's OEIS frontend

A006481 Euler characteristics of polytopes.

1, 2, 3, 4, 5, 11, 21, 36, 57, 127, 253, 463, 793, 1717, 3433, 6436, 11441, 24311, 48621, 92379, 167961, 352717, 705433, 1352079, 2496145, 5200301, 10400601, 20058301, 37442161

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
H. Hadwiger and P. Mani, On polyhedra with extremal Euler characteristic, J. Combin. Theory, A 17 (1974), 345-349. See p. 346.

Very like A051920. Cf. A320996.

a[n_] := Binomial[n-1, 2*Floor[(n-1)/4] + 1] + 1; Table[a[n], {n, 1, 29}] (* Jean-François Alcover, Jan 23 2012, after Ralf Stephan *)

Numbers suggest that for n not divisible by 4, a(n) = C(n, [n/2]) + 1 and C(n, [(n-1)/2]) + 1 otherwise (see A051920 and A037952+1). - Ralf Stephan, Jun 07 2005

A116399 Triangle whose k-th column has e.g.f. sum{j=0..k, Bessel_I(k+j,2x)}.

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 0, 3, 1, 1, 6, 4, 5, 1, 1, 0, 10, 5, 6, 1, 1, 20, 15, 21, 7, 7, 1, 1, 0, 35, 21, 28, 8, 8, 1, 1, 70, 56, 84, 36, 37, 9, 9, 1, 1, 0, 126, 84, 120, 45, 46, 10, 10, 1, 1, 252, 210, 330, 165, 175, 56, 56, 11, 11, 1, 1

Offset: 0

Paul Barry, Feb 13 2006

Second column is A037952. Third column is the double of A002054. Product of A007318 and A116399 is A116401.

			Triangle begins
1,
0, 1,
2, 1, 1,
0, 3, 1, 1,
6, 4, 5, 1, 1,
0, 10, 5, 6, 1, 1,
20, 15, 21, 7, 7, 1, 1

A143359 Triangle read by rows, T(n,k) = number of symmetric ordered trees with n edges and root degree k (1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 1, 1, 1, 6, 0, 3, 0, 1, 10, 2, 4, 2, 1, 1, 20, 0, 10, 0, 4, 0, 1, 35, 5, 15, 5, 5, 3, 1, 1, 70, 0, 35, 0, 15, 0, 5, 0, 1, 126, 14, 56, 14, 21, 9, 6, 4, 1, 1, 252, 0, 126, 0, 56, 0, 21, 0, 6, 0, 1, 462, 42, 210, 42, 84, 28, 28, 14, 7, 5, 1, 1, 924, 0, 462, 0, 210, 0, 84, 0, 28, 0, 7, 0, 1

Offset: 1

Emeric Deutsch, Aug 15 2008

Number of symmetric Dyck n-paths with k returns to the x-axis. - David Scambler, Aug 16 2012

			Triangle starts:
   1;
   1,  1;
   2,  0,  1;
   3,  1,  1,  1;
   6,  0,  3,  0,  1;
  10,  2,  4,  2,  1,  1;
  20,  0, 10,  0,  4,  0,  1;
  35,  5, 15,  5,  5,  3,  1,  1;

Cf. A001405, A000108 (column 2), A143360, A037952 (column 3).

C:=proc(z) options operator, arrow: (1/2-(1/2)*sqrt(1-4*z))/z end proc: S:=1/(1-z-z^2*C(z^2)): G:=(1+t*z*S)/(1-t^2*z^2*C(z^2))-1: Gser:=simplify(series(G, z=0,15)): for n to 13 do P[n]:=coeff(Gser,z,n) end do: for n to 13 do seq(coeff(P[n],t,j),j=1..n) end do; # yields sequence in triangular form

Module[{nmax = 13, G, C, S},
   G = (1 + t*z*S[z])/(1 - t^2*z^2*C[z^2]) - 1;
   S[z_] = 1/(1 - z - z^2*C[z^2]) ;
   C[z_] = (1 - Sqrt[1 - 4 z])/(2 z);
   CoefficientList[#/t + O[t]^nmax, t]& /@
   CoefficientList[G/z + O[z]^nmax, z]
] // Flatten (* Jean-François Alcover, Apr 09 2024 *)

G.f. = (1+t*z*S)/(1-t^2*z^2*C(z^2))-1, where S = 1/(1-z-z^2*C(z^2)) is the g.f. of the sequence binomial(n, floor(n/2)) (A001405) and C(z) = (1-sqrt(1-4z))/(2z) is the generating function of the Catalan numbers (A000108).
Sum_{k=1..n} T(n,k) = A001405(n).
T(n,1) = A001405(n-1).
Sum_{k=1..n} k*T(n,k) = A143360(n).
Sum_{k=2..n} T(n,k) = A037952(n). - R. J. Mathar, Sep 24 2021

A369589 Triangular array read by rows: T(m,n) = number of Yamanouchi words over the alphabet {1, 2, ..., n} of length m that start with n, m >= 1, n = 1..m.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 4, 5, 3, 1, 1, 10, 14, 11, 4, 1, 1, 15, 35, 31, 19, 5, 1, 1, 35, 90, 106, 69, 29, 6, 1, 1, 56, 245, 323, 265, 127, 41, 7, 1, 1, 126, 644, 1093, 971, 583, 209, 55, 8, 1, 1, 210, 1716, 3439, 3644, 2446, 1106, 319, 71, 9, 1, 1, 462, 4707, 11716, 13771, 10616, 5323, 1904, 461, 89, 10, 1

Offset: 1

Max Alekseyev, Jan 26 2024

			Array starts with
  m=1: 1
  m=2: 1,  1
  m=3: 1,  1,  1
  m=4: 1,  3,  2,  1
  m=5: 1,  4,  5,  3,  1
  m=6: 1, 10, 14, 11,  4, 1
  m=7: 1, 15, 35, 31, 19, 5, 1

Wikipedia, Lattice word.

Row sums: A238728.
Columns: A000012 (n=1), A037952 (n=2), A369591 (n=3).
Cf. A001006, A369588.

A378060 a(n) = binomial(n, floor((n-1)/2))^2.

Original entry on oeis.org

0, 1, 1, 9, 16, 100, 225, 1225, 3136, 15876, 44100, 213444, 627264, 2944656, 9018009, 41409225, 130873600, 590976100, 1914762564, 8533694884, 28210561600, 124408576656, 418151049316, 1828114918084, 6230734868736, 27043120090000, 93271169290000, 402335398890000

Offset: 0

Peter Luschny, Dec 03 2024

Number of walks of length n with unit steps in all four directions (NSWE), starting at the origin and ending on the y-axis, never going below the x-axis and the end point having a positive height.

			The 16 walks of length 4: NNNN, NNNS, NNSN, NNEW, NNWE, NSNN, NENW, NEWN, NWNE, NWEN, ENNW, ENWN, EWNN, WNNE, WNEN, WENN.

Cf. A060150 (odd bisection), A337900 (even bisection), A037952, A378061.

# Generates the walks (for illustration only).
function aCount(n::Int)
    a = [""]
    c = 0
    for w in a
        if length(w) == n
            if (count('N', w) != count('S', w) && count('W', w) == count('E', w))
                c += 1
                # println(w)
            end
        else
            for j in "NSEW"
                u = string(w, j)
                if count('N', u) >= count('S', u)
                   push!(a, u)
    end end end end
    return c
end
println([aCount(n) for n in 0:11])

a := n -> binomial(n, iquo(n+1, 2) - 1)^2: seq(a(n), n = 0..27);
a := proc(n) option remember; if n < 2 then n else ((32*n^5 - 48*n^4 + 16*n^2)*a(n - 2) + (8*n^4 - 20*n^2)*a(n - 1))/(2*(n - 1)^2*(n - 1/2)*(n + 2)^2) fi end:
# Alternative:
egf := BesselI(0, 2*x)^2 + BesselI(1, 2*x)*BesselI(0, 2*x)*(1 - 1/x):
ser := series(egf, x, 29): seq(n!*coeff(ser, x, n), n = 0..27);

Array[Binomial[#, Floor[(# + 1)/2] - 1]^2 &, 28, 0] (* Michael De Vlieger, Dec 04 2024 *)

a(n) = n!*[x^n] (BesselI(0, 2*x)^2 + BesselI(1, 2*x)*BesselI(0, 2*x)*(1 - 1/x)).
a(n) = [x^n] (((8*x^2 + 2*x)*EllipticK(4*x) - Pi*(1 + x) + 2*EllipticE(4*x))/(4*x^2*Pi)).
a(n) = [x^n] (x*hypergeom([1,3/2,3/2], [2,2], 16*x^2) + x^2*hypergeom([3/2,3/2,2,2], [1,3,3], 16*x^2)).
a(n) = Sum_{k=0..n} (-1)^(n-k+N)*C(n-k, N)*C(n, k)*C(n+k, k), where N = floor((n-1)/2) and C = binomial.
Recurrence: a(n) = ((32*n^5 - 48*n^4 + 16*n^2)*a(n - 2) + (8*n^4 - 20*n^2)*a(n - 1))/(2*(n - 1)^2*(n - 1/2)*(n + 2)^2).
a(n) = Sum_{k=1..n} A378061(n, k).

A056641 Least positive integer k for which (b+1)^k is not palindromic in base b, b = 2, 3, 4, ...

Original entry on oeis.org

4, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 2

Helge Robitzsch (hrobi(AT)math.uni-goettingen.de), Aug 11 2000

Sequence of run lengths is C(n,[ (n-1)/2 ]) (= A037952), n=1,2,3,...; sequence of b where a(b) != a(b-1), b >= 3, is C(b-1,[ (b-1)/2 ]) (= A001405).

			The 4th term is 4 because base 5 representations of (5+1)^1 = 11, (5+1)^2 = 121, (5+1)^3 = 1331, are all palindromic, while (5+1)^4 = 20141 is not.

Cf. A037952, A001405, A305233.

palq[x_] := x == Reverse[x] Table[x = 0; While[palq[IntegerDigits[(t + 1)^x, t]], ++x]; x, {t, START, FINISH}] (* Dylan Hamilton, Aug 15 2010 *)

A191525 Triangle read by rows: T(n,k) is the number of left factors of Dyck paths of length n and having k hills; a hill is a (1,1)-step starting at level 0 and followed by a (1,-1)-step.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 1, 1, 7, 2, 1, 13, 5, 1, 1, 24, 8, 2, 1, 46, 16, 6, 1, 1, 86, 28, 9, 2, 1, 166, 58, 19, 7, 1, 1, 314, 103, 32, 10, 2, 1, 610, 211, 71, 22, 8, 1, 1, 1163, 382, 121, 36, 11, 2, 1, 2269, 781, 261, 85, 25, 9, 1, 1, 4352, 1432, 456, 140, 40, 12, 2, 1, 8518, 2920, 976, 316, 100, 28, 10, 1, 1

Offset: 0

Emeric Deutsch, Jun 06 2011

Row n contains 1+floor(n/2) entries.
Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
T(n,0)=A191526(n).
Sum(k*T(n,k), k>=0) = A037952(n-1).

			T(5,1)=2 because we have (UD)UUD and (UD)UUU, where U=(1,1) and D=(1,-1) (the hills are shown between parentheses).
Triangle starts:
  1;
  1;
  1,1;
  2,1;
  4,1,1;
  7,2,1;
  13,5,1,1;

Cf. A001405, A191526, A037952, A065600.

c := ((1-sqrt(1-4*z^2))*1/2)/z^2: G := 1/((1-z*c)*(1-z^4*c^2-t*z^2)): Gser := simplify(series(G, z = 0, 19)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

G.f.: G(t,z)=1/((1-z*c)*(1-z^4*c^2-t*z^2)), where c=(1-sqrt(1-4*z^2))/(2*z^2) is the Catalan function with argument z^2.

A335322 Triangle read by rows: T(n, k) = binomial(n, floor((n+k+1)/2)) with k <= n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 10, 5, 5, 1, 1, 15, 15, 6, 6, 1, 1, 35, 21, 21, 7, 7, 1, 1, 56, 56, 28, 28, 8, 8, 1, 1, 126, 84, 84, 36, 36, 9, 9, 1, 1, 210, 210, 120, 120, 45, 45, 10, 10, 1, 1, 462, 330, 330, 165, 165, 55, 55, 11, 11, 1, 1, 792, 792, 495, 495, 220, 220, 66, 66, 12, 12, 1, 1

Offset: 1

Stefano Spezia, May 31 2020

T(n, k) is a tight upper bound of the cardinality of an intersecting Sperner family or antichain of the set {1, 2,..., n}, where every collection of pairwise independent subsets is characterized by an intersection of cardinality at least k (see Theorem 1.3 in Wong and Tay).

Equals A061554 with the first row of the array (resp. the first column of the triangle) removed. - Georg Fischer, Jul 26 2023

			The triangle T(n, k) begins
n\k|  1   2   3   4   5   6   7   8
---+-------------------------------
1  |  1
2  |  1   1
3  |  3   1   1
4  |  4   4   1   1
5  | 10   5   5   1   1
6  | 15  15   6   6   1   1
7  | 35  21  21   7   7   1   1
8  | 56  56  28  28   8   8   1   1
...

Eric Charles Milner, A Combinatorial Theorem On Systems of Sets, Journal of the London Mathematical Society, 43, (1968), 204-206.
W. H. W. Wong and E. G. Tay, On Cross-intersecting Sperner Families, arXiv:2001.01910 [math.CO], 2020.
Index entries for sequences related to antichains.

Cf. A000372, A001405, A004526, A007318, A007695, A061554, A266696, A325982, A325983.

Cf. A037951 (k=3), A037952 (k=1), A037953 (k=5), A037954 (k=7), A037955 (k=2), A037956 (k=4), A037957 (k=6), A037958 (k=8), A045621 (row sums).

T[n_,k_]:=Binomial[n,Floor[(n+k+1)/2]]; Table[T[n,k],{n,12},{k,n}]//Flatten

T(n, k) = binomial(n, (n+k+1)\2);
vector(10, n, vector(n, k, T(n, k))) \\ Michel Marcus, Jun 01 2020

T(n, k) = A007318(n, A004526(n+k+1)) with k <= n.

A348013 Triangle by rows: T(n,k) is the number of n-step Dyck paths with k catastrophes.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 4, 7, 3, 1, 10, 14, 12, 4, 1, 15, 37, 31, 18, 5, 1, 35, 74, 90, 56, 25, 6, 1, 56, 176, 216, 179, 90, 33, 7, 1, 126, 352, 552, 492, 315, 134, 42, 8, 1, 210, 794, 1269, 1362, 966, 510, 189, 52, 9, 1, 462, 1588, 3033, 3480, 2890, 1716, 777, 256, 63, 10, 1, 792, 3473

Offset: 1

R. J. Mathar, Sep 24 2021

T(n,k) is the number chains of k "incomplete" Dyck paths with a total length of n. (Incomplete Dyck paths are those not ending at the horizontal axis.) Each of the k subsections of the paths does not return to the horizontal axis; they are commonly referred to as paths with catastrophes (like black Fridays on stock market charts).

			The triangle starts
    1
    1    1
    3    2    1
    4    7    3    1
   10   14   12    4    1
   15   37   31   18    5    1
   35   74   90   56   25    6    1
   56  176  216  179   90   33    7    1
  126  352  552  492  315  134   42    8    1
  210  794 1269 1362  966  510  189   52    9    1
  462 1588 3033 3480 2890 1716  777  256   63   10    1
  792 3473 6781 8901 8060 5521 2835 1130  336   75   11    1
T(1,1)=1 counts U| where the vertical bar indicates starting a new path at the horizontal axis (the catastrophe).
T(2,1)=1 counts UU|.
T(4,1)=4 counts UUUU|, UUUD|, UUDU|, UDUU|.
T(3,2)=2 counts UU|U| and U|UU| .
T(4,2)=7 counts U|UUU|, U|UUD|, U|UDU|, UU|UU|, UUU|U|, UUD|U| and UDU|U|.

Cf. A348012 (row sums), A037952 (k=1), A191389 (k=2).

T(n,1) = A037952(n).
T(n,2) = A191389(n+2).
The generating function of column k is g037952(x)^k, where g037952(x) = x +x^2 +3*x^3+... is the generating function of A037952.

cp's OEIS Frontend

A006481 Euler characteristics of polytopes.

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A116399 Triangle whose k-th column has e.g.f. sum{j=0..k, Bessel_I(k+j,2x)}.

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A143359 Triangle read by rows, T(n,k) = number of symmetric ordered trees with n edges and root degree k (1 <= k <= n).

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A369589 Triangular array read by rows: T(m,n) = number of Yamanouchi words over the alphabet {1, 2, ..., n} of length m that start with n, m >= 1, n = 1..m.

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A378060 a(n) = binomial(n, floor((n-1)/2))^2.

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A056641 Least positive integer k for which (b+1)^k is not palindromic in base b, b = 2, 3, 4, ...

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A191525 Triangle read by rows: T(n,k) is the number of left factors of Dyck paths of length n and having k hills; a hill is a (1,1)-step starting at level 0 and followed by a (1,-1)-step.

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A335322 Triangle read by rows: T(n, k) = binomial(n, floor((n+k+1)/2)) with k <= n.

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A348013 Triangle by rows: T(n,k) is the number of n-step Dyck paths with k catastrophes.

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