A105937 -id:A105937 - cp's OEIS frontend

A127080 Infinite square array read by antidiagonals: Q(m, 0) = 1, Q(m, 1) = 1; Q(m, 2k) = (m - 2k + 1)Q(m+1, 2k-1) - (2k-1)Q(m+2,2k-2), mQ(m, 2k+1) = (m - 2k)Q(m+1, 2k) - 2k(m+1)*Q(m+2, 2k-1).

1, 1, 1, 1, 1, -2, 1, 1, -1, -5, 1, 1, 0, -4, 12, 1, 1, 1, -3, 3, 43, 1, 1, 2, -2, -4, 28, -120, 1, 1, 3, -1, -9, 15, -15, -531, 1, 1, 4, 0, -12, 4, 48, -288, 1680, 1, 1, 5, 1, -13, -5, 75, -105, 105, 8601, 1, 1, 6, 2, -12, -12, 72, 24, -624, 3984, -30240, 1, 1, 7, 3, -9, -17, 45, 105, -735, 945, -945, -172965

Offset: 0

N. J. A. Sloane, Mar 24 2007

Comment from N. J. A. Sloane, Jan 29 2020: (Start)
It looks like there was a missing 2 in the definition, which I have now corrected.  The old definition was:
(Wrong!) Infinite square array read by antidiagonals: Q(m, 0) = 1, Q(m, 1) = 1; Q(m, 2k) = (m - 2k + 1)*Q(m+1, 2k-1) - (2k-1)*Q(m+2, k-2), m*Q(m, 2k+1) = (m - 2k)*Q(m+1, 2k) - 2k(m+1)*Q(m+2, 2k-1). (Wrong!) (End)

			Array begins:
     1,    1,    1,    1,    1,   1,   1,    1,    1,    1, ... (A000012)
     1,    1,    1,    1,    1,   1,   1,    1,    1,    1, ... (A000012)
    -2,   -1,    0,    1,    2,   3,   4,    5,    6,    7, ... (A023444)
    -5,   -4,   -3,   -2,   -1,   0,   1,    2,    3,    4, ... (A023447)
    12,    3,   -4,   -9,  -12, -13, -12,   -9,   -4,    3, ... (A127146)
    43,   28,   15,    4,   -5, -12, -17,  -20,  -21,  -20, ... (A127147)
  -120,  -15,   48,   75,   72,  45,   0,  -57, -120, -183, ... (A127148)
  -531, -288, -105,   24,  105, 144, 147,  120,   69,    0, ...
  1680,  105, -624, -735, -432, 105, 720, 1281, 1680, 1833, ...

V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.

G. C. Greubel, Antidiagonals n = 0..100, flattened

See A105937 for another version.
Columns give A127137, A127138, A127144, A127145;
Rows give A127146, A127147, A127148.

f:= proc(k) option remember;
      if `mod`(k,2)=0 then k!/(k/2)!
    else 2^(k-1)*((k-1)/2)!*add(binomial(2*j, j)/8^j, j=0..((k-1)/2))
      fi; end;
Q:= proc(n, k) option remember;
      if n=0 then (-1)^binomial(k, 2)*f(k)
    elif k<2 then 1
    elif `mod`(k,2)=0 then (n-k+1)*Q(n+1,k-1) - (k-1)*Q(n+2,k-2)
    else ( (n-k+1)*Q(n+1,k-1) - (k-1)*(n+1)*Q(n+2,k-2) )/n
      fi; end;
seq(seq(Q(n-k, k), k=0..n), n=0..12); # G. C. Greubel, Jan 30 2020

Q[0, k_]:= Q[0,k]= (-1)^Binomial[k, 2]*If[EvenQ[k], k!/(k/2)!, 2^(k-1)*((k-1)/2)!* Sum[Binomial[2*j, j]/8^j, {j, 0, (k-1)/2}] ];
Q[n_, k_]:= Q[n,k]= If[k<2, 1, If[EvenQ[k], (n-k+1)*Q[n+1, k-1] - (k-1)*Q[n+2, k-2], ((n -k+1)*Q[n+1, k-1] - (k-1)*(n+1)*Q[n+2, k-2])/n]];
Table[Q[n-k,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 30 2020 *)

@CachedFunction
def f(k):
    if (mod(k, 2)==0): return factorial(k)/factorial(k/2)
    else: return 2^(k-1)*factorial((k-1)/2)*sum(binomial(2*j, j)/8^j for j in (0..(k-1)/2))
def Q(n,k):
    if (n==0): return (-1)^binomial(k, 2)*f(k)
    elif (k<2): return 1
    elif (mod(k,2)==0): return (n-k+1)*Q(n+1,k-1) - (k-1)*Q(n+2,k-2)
    else: return ( (n-k+1)*Q(n+1,k-1) - (k-1)*(n+1)*Q(n+2,k-2) )/n
[[Q(n-k,k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 30 2020

E.g.f.: Sum_{k >= 0} Q(m,2k) x^k/k! = (1+4x)^((m-1)/2)/(1+2x)^(m/2), Sum_{k >= 0} Q(m,2k+1) x^k/k! = (1+4x)^((m-2)/2)/(1+2x)^((m+1)/2).

More terms added by G. C. Greubel, Jan 30 2020

A126934 Define an array by d(m, 0) = 1, d(m, 1) = m; d(m, k) = (m - k + 1) d(m+1, k-1) - (k-1) (m+1) d(m+2, k-2). Sequence gives d(0,2n).

Original entry on oeis.org

1, -2, 36, -1800, 176400, -28576800, 6915585600, -2337467932800, 1051860569760000, -607975409321280000, 438958245529964160000, -387161172557428389120000, 409616520565759235688960000, -512020650707199044611200000000, 746526108731096207043129600000000, -1255656914885703820246543987200000000

Offset: 0

Vincent v.d. Noort, Mar 21 2007

sign

|a(n)| is the number of functions f:{1,2,...,2n}->{1,2,...,2n} such that each element has either 0 or 2 preimages. That is, |(f^-1)(x)| is in {0,2} for all x in {1,2,...,2n}. - Geoffrey Critzer, Feb 24 2012.

V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.

G. C. Greubel, Table of n, a(n) for n = 0..150
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 131.
S. Goodenough, C. Lavault, On subsets of Riordan subgroups and Heisenberg--Weyl algebra, arXiv preprint arXiv:1404.1894 [cs.DM], 2014-2016.
S. Goodenough, C. Lavault, Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups, The Electronic Journal of Combinatorics, 22(4) (2015), #P4.16.

See A105937 for the full array.

A126935 Define an array by d(m, 0) = 1, d(m, 1) = m; d(m, k) = (m - k + 1) d(m+1, k-1) - (k-1) (m+1) d(m+2, k-2). Sequence gives d(n,3).

Original entry on oeis.org

0, -12, -24, -30, -24, 0, 48, 126, 240, 396, 600, 858, 1176, 1560, 2016, 2550, 3168, 3876, 4680, 5586, 6600, 7728, 8976, 10350, 11856, 13500, 15288, 17226, 19320, 21576, 24000, 26598, 29376, 32340, 35496, 38850, 42408, 46176, 50160, 54366, 58800, 63468, 68376, 73530

Offset: 0

Vincent v.d. Noort, Mar 21 2007

V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

A row of A105937.

[n*(n+2)*(n-5): n in [0..50]]; // G. C. Greubel, Jan 29 2020

seq( n*(n+2)*(n-5), n=0..50); # G. C. Greubel, Jan 29 2020

T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==1, n, (n-k+1)*T[n+1, k-1] - (k-1)*(n+1)* T[n+2, k-2]]]; Table[T[n, 3], {n,0,50}] (* G. C. Greubel, Jan 29 2020 *)

vector(50, n, my(m=n-1); m*(m+2)*(m-5) ) \\ G. C. Greubel, Jan 29 2020

[n*(n+2)*(n-5) for n in (0..50)] # G. C. Greubel, Jan 29 2020

a(n) = n*(n+2)*(n-5).
From G. C. Greubel, Jan 29 2020: (Start)
G.f.: (-6)*x*(2 - 4*x + x^2)/(1-x)^4.
E.g.f.: x*(-12 + x^2)*exp(x). (End)

A126958 Define an array by d(m, 0) = 1, d(m, 1) = m; d(m, k) = (m - k + 1) d(m+1, k-1) - (k-1) (m+1) d(m+2, k-2). Sequence gives d(n,4).

Original entry on oeis.org

36, 24, -60, -216, -420, -624, -756, -720, -396, 360, 1716, 3864, 7020, 11424, 17340, 25056, 34884, 47160, 62244, 80520, 102396, 128304, 158700, 194064, 234900, 281736, 335124, 395640, 463884, 540480, 626076, 721344, 826980, 943704, 1072260, 1213416, 1367964

Offset: 0

Vincent v.d. Noort, Mar 21 2007

V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

A row of A105937.

List([0..40], n-> (n+3)*(n+1)*(n^2 -10*n +12)); # G. C. Greubel, Jan 29 2020

[(n+3)*(n+1)*(n^2 -10*n +12): n in [0..40]]; // G. C. Greubel, Jan 29 2020

seq( (n+3)*(n+1)*(n^2 -10*n +12), n=0..40); # G. C. Greubel, Jan 29 2020

Table[(n+3)*(n+1)*(n^2 -10*n +12), {n,0,40}] (* G. C. Greubel, Jan 29 2020 *)

vector(41, n, my(m=n-1); (m+3)*(m+1)*(m^2 -10*m +12)) \\ G. C. Greubel, Jan 29 2020

[(n+3)*(n+1)*(n^2 -10*n +12) for n in (0..40)] # G. C. Greubel, Jan 29 2020

a(n) = (n+3)*(n+1)*(n^2 -10*n +12).
From G. C. Greubel, Jan 29 2020: (Start)
G.f.: 12*(3 -13*x +15*x^2 -3*x^3)/(1-x)^5.
E.g.f.: (36 -12*x -36*x^2 +x^4)*exp(x). (End)

A126962 Define an array by d(m, 0) = 1, d(m, 1) = m; d(m, k) = (m - k + 1) d(m+1, k-1) - (k-1) (m+1) d(m+2, k-2). Sequence gives d(1,n).

Original entry on oeis.org

1, 1, -2, -12, 24, 420, -720, -30240, 40320, 3764880, -3628800, -728481600, 479001600, 203545742400, -87178291200, -77806624896000, 20922789888000, 39045031657632000, -6402373705728000, -24904933604014464000, 2432902008176640000, 19678195269815322240000, -1124000727777607680000

Offset: 0

Vincent v.d. Noort, Mar 21 2007

sign

V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.

G. C. Greubel, Table of n, a(n) for n = 0..150

A column of A105937.

function T(n,k)
  if k eq 0 then return 1;
  elif k eq 1 then return n;
  else return (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2);
  end if; return T; end function;
[T(1,n): n in [0..25]]; // G. C. Greubel, Jan 29 2020

T:= proc(n, k) option remember;
      if k=0 then 1
    elif k=1 then n
    else (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2)
      fi; end:
seq(T(1, n), n=0..25); # G. C. Greubel, Jan 29 2020

T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==1, n, (n-k+1)*T[n+1, k-1] - (k-1)*(n+1)* T[n+2, k-2]]]; Table[T[1, n], {n,0,25}] (* G. C. Greubel, Jan 29 2020 *)

T(n,k) = if(k==0, 1, if(k==1, n, (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2) ));
vector(25, n, T(1, (n-1)) ) \\ G. C. Greubel, Jan 29 2020

@CachedFunction
def T(n, k):
    if (k==0): return 1
    elif (k==1): return n
    else: return (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2)
[T(1, n) for n in (0..25)] # G. C. Greubel, Jan 29 2020

A127067 Define an array by d(m, 0) = 1, d(m, 1) = m; d(m, k) = (m - k + 1) d(m+1, k-1) - (k-1) (m+1) d(m+2, k-2). Sequence gives d(2,n).

Original entry on oeis.org

1, 2, 0, -24, -60, 720, 5040, -40320, -589680, 3628800, 99792000, -479001600, -23740516800, 87178291200, 7682586912000, -20922789888000, -3281772285792000, 6402373705728000, 1801868049805824000, -2432902008176640000, -1241948957556827520000, 1124000727777607680000

Offset: 0

Vincent v.d. Noort, Mar 21 2007

sign

V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.

G. C. Greubel, Table of n, a(n) for n = 0..350

A column of A105937.

function T(n, k)
  if k eq 0 then return 1;
  elif k eq 1 then return n;
  else return (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2);
  end if; return T; end function;
[T(2, n): n in [0..25]]; // G. C. Greubel, Jan 30 2020

T:= proc(n, k) option remember;
      if k=0 then 1
    elif k=1 then n
    else (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2)
      fi; end:
seq(T(2, n), n=0..25); # G. C. Greubel, Jan 30 2020

T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==1, n, (n-k+1)*T[n+1, k-1] - (k-1)*(n+1)* T[n+2, k-2]]]; Table[T[2, n], {n,0,25}] (* G. C. Greubel, Jan 30 2020 *)

T(n, k) = if(k==0, 1, if(k==1, n, (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2) ));
vector(25, n, T(2, n-1) ) \\ G. C. Greubel, Jan 30 2020

@CachedFunction
def T(n, k):
    if (k==0): return 1
    elif (k==1): return n
    else: return (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2)
[T(2, n) for n in (0..25)] # G. C. Greubel, Jan 30 2020

A127068 Let d(m, 0) = 1, d(m, 1) = m, and d(m, k) = (m - k + 1)d(m+1, k-1) - (k-1)(m+1) d(m+2, k-2). Sequence gives d(3,n).

Original entry on oeis.org

1, 3, 4, -30, -216, 420, 14400, 22680, -1411200, -8482320, 195955200, 2399997600, -36883123200, -788107320000, 9066542284800, 318173519664000, -2824576634880000, -159078423407904000, 1088403529973760000, 97970873094110016000, -508476519708917760000, -73631427647097640320000

Offset: 0

Vincent v.d. Noort, Mar 21 2007

sign

V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.

G. C. Greubel, Table of n, a(n) for n = 0..150

A column of A105937.

function T(n, k)
  if k eq 0 then return 1;
  elif k eq 1 then return n;
  else return (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2);
  end if; return T; end function;
[T(3, n): n in [0..25]]; // G. C. Greubel, Jan 29 2020

T:= proc(n, k) option remember;
      if k=0 then 1
    elif k=1 then n
    else (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2)
      fi; end:
seq(T(3, n), n=0..25); # G. C. Greubel, Jan 29 2020

T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==1, n, (n-k+1)*T[n+1, k-1] - (k-1)*(n+1)* T[n+2, k-2]]]; Table[T[3, n], {n,0,25}] (* G. C. Greubel, Jan 29 2020 *)

T(n, k) = if(k==0, 1, if(k==1, n, (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2) ));
vector(25, n, T(3, (n-1)) ) \\ G. C. Greubel, Jan 29 2020

@CachedFunction
def T(n, k):
    if (k==0): return 1
    elif (k==1): return n
    else: return (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2)
[T(3, n) for n in (0..25)] # G. C. Greubel, Jan 29 2020

From Peter Bala, Feb 15 2022: (Start)
Conjectures:
a(2*n) = (-1)^(n+1)*(n + 1)*(2*n - 1)*(2*n)!.
a(2*n+1) = - 2*(2*n + 3)*(3*n - 2)*a(2*n-1) - 4*(n - 1)*(2*n + 3)*(4*n^2 - 1)*a(2*n-3) with a(1) = 3 and a(3) = -30. (End)

A127070 Define an array by d(m, 0) = 1, d(m, 1) = m; d(m, k) = (m - k + 1) d(m+1, k-1) - (k-1) (m+1) d(m+2, k-2). Sequence gives d(4,n).

Original entry on oeis.org

1, 4, 10, -24, -420, -960, 22680, 201600, -1496880, -36288000, 64864800, 7823692800, 25297272000, -2092278988800, -18988521552000, 690452066304000, 11457025515936000, -277436193914880000, -7430805000755136000, 133809610449715200000, 5500591866494524800000, -76432049488877322240000

Offset: 0

Vincent v.d. Noort, Mar 21 2007

sign

V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.

G. C. Greubel, Table of n, a(n) for n = 0..150

A column of A105937.

function T(n, k)
  if k eq 0 then return 1;
  elif k eq 1 then return n;
  else return (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2);
  end if; return T; end function;
[T(4, n): n in [0..25]]; // G. C. Greubel, Jan 29 2020

T:= proc(n, k) option remember;
      if k=0 then 1
    elif k=1 then n
    else (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2)
      fi; end:
seq(T(4, n), n=0..25); # G. C. Greubel, Jan 29 2020

T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==1, n, (n-k+1)*T[n+1, k-1] - (k-1)*(n+1)* T[n+2, k-2]]]; Table[T[4, n], {n,0,25}] (* G. C. Greubel, Jan 29 2020 *)

T(n, k) = if(k==0, 1, if(k==1, n, (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2) ));
vector(25, n, T(4, (n-1)) ) \\ G. C. Greubel, Jan 29 2020

@CachedFunction
def T(n, k):
    if (k==0): return 1
    elif (k==1): return n
    else: return (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2)
[T(4, n) for n in (0..25)] # G. C. Greubel, Jan 29 2020

cp's OEIS Frontend

A127080 Infinite square array read by antidiagonals: Q(m, 0) = 1, Q(m, 1) = 1; Q(m, 2k) = (m - 2k + 1)*Q(m+1, 2k-1) - (2k-1)*Q(m+2,2k-2), m*Q(m, 2k+1) = (m - 2k)*Q(m+1, 2k) - 2k(m+1)*Q(m+2, 2k-1).

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A126934 Define an array by d(m, 0) = 1, d(m, 1) = m; d(m, k) = (m - k + 1) d(m+1, k-1) - (k-1) (m+1) d(m+2, k-2). Sequence gives d(0,2n).

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A126935 Define an array by d(m, 0) = 1, d(m, 1) = m; d(m, k) = (m - k + 1) d(m+1, k-1) - (k-1) (m+1) d(m+2, k-2). Sequence gives d(n,3).

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A126958 Define an array by d(m, 0) = 1, d(m, 1) = m; d(m, k) = (m - k + 1) d(m+1, k-1) - (k-1) (m+1) d(m+2, k-2). Sequence gives d(n,4).

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A126962 Define an array by d(m, 0) = 1, d(m, 1) = m; d(m, k) = (m - k + 1) d(m+1, k-1) - (k-1) (m+1) d(m+2, k-2). Sequence gives d(1,n).

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Magma

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Mathematica

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A127067 Define an array by d(m, 0) = 1, d(m, 1) = m; d(m, k) = (m - k + 1) d(m+1, k-1) - (k-1) (m+1) d(m+2, k-2). Sequence gives d(2,n).

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A127080 Infinite square array read by antidiagonals: Q(m, 0) = 1, Q(m, 1) = 1; Q(m, 2k) = (m - 2k + 1)Q(m+1, 2k-1) - (2k-1)Q(m+2,2k-2), mQ(m, 2k+1) = (m - 2k)Q(m+1, 2k) - 2k(m+1)*Q(m+2, 2k-1).

A127068 Let d(m, 0) = 1, d(m, 1) = m, and d(m, k) = (m - k + 1)d(m+1, k-1) - (k-1)(m+1) d(m+2, k-2). Sequence gives d(3,n).