A127080 -id:A127080 - cp's OEIS frontend

A127138 Q(1,n), where Q(m,k) is defined in A127080 and A127137.

1, 1, -1, -4, 3, 28, -15, -288, 105, 3984, -945, -70080, 10395, 1506240, -135135, -38384640, 2027025, 1133072640, -34459425, -38038533120, 654729075, 1431213235200, -13749310575, -59645279232000, 316234143225, 2726781752217600, -7905853580625, -135661078090137600, 213458046676875

Offset: 0

N. J. A. Sloane, Mar 24 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..500

A001147 interleaved with A076729.
Column 1 of A127080.
Cf. A127137, A127144, A127145.

Q:= proc(n, k) option remember;
      if 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( Q(1, n), n=0..30); # G. C. Greubel, Jan 30 2020

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[1, k], {k,0,30}] (* G. C. Greubel, Jan 30 2020 *)

@CachedFunction
def Q(n,k):
    if (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(1,n) for n in (0..30)] # G. C. Greubel, Jan 30 2020

See A127080 for e.g.f.

A127144 Q(2,n), where Q(m,k) is defined in A127080 and A127137.

Original entry on oeis.org

1, 1, 0, -3, -4, 15, 48, -105, -624, 945, 9600, -10395, -175680, 135135, 3790080, -2027025, -95235840, 34459425, 2752081920, -654729075, -90328089600, 13749310575, 3328103116800, -316234143225, -136191650918400, 7905853580625, 6131573025177600, -213458046676875, -301213549769932800

Offset: 0

N. J. A. Sloane, Mar 24 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..500

A126967 interleaved with A001147.
Column 2 of A127080.
Cf. A127137, A127138, A127145.

Q:= proc(n, k) option remember;
      if 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( Q(2, n), n=0..30); # G. C. Greubel, Jan 30 2020

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[2, k], {k,0,30}] (* G. C. Greubel, Jan 30 2020 *)

@CachedFunction
def Q(n,k):
    if (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(2,n) for n in (0..30)] # G. C. Greubel, Jan 30 2020

See A127080 for e.g.f.

A127145 Q(3,n), where Q(m,k) is defined in A127080 and A127137.

Original entry on oeis.org

1, 1, 1, -2, -9, 4, 75, 24, -735, -816, 8505, 17760, -114345, -388800, 1756755, 9233280, -30405375, -242968320, 585810225, 7125511680, -12439852425, -232838323200, 288735522075, 8450546227200, -7273385294175, -339004760371200, 197646339515625, 14945696794828800, -5763367260275625

Offset: 0

N. J. A. Sloane, Mar 24 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..500

Cf. A126965.
Column 3 of A127080.
Cf. A127137, A127138, A127144.

Q:= proc(n, k) option remember;
      if 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( Q(3, n), n=0..30); # G. C. Greubel, Jan 30 2020

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[3, k], {k,0,30}] (* G. C. Greubel, Jan 30 2020 *)

@CachedFunction
def Q(n,k):
    if (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(3,n) for n in (0..30)] # G. C. Greubel, Jan 30 2020

See A127080 for e.g.f.

A127147 Q(n,5), where Q(m,k) is defined in A127080 and A127137.

Original entry on oeis.org

43, 28, 15, 4, -5, -12, -17, -20, -21, -20, -17, -12, -5, 4, 15, 28, 43, 60, 79, 100, 123, 148, 175, 204, 235, 268, 303, 340, 379, 420, 463, 508, 555, 604, 655, 708, 763, 820, 879, 940, 1003, 1068, 1135, 1204, 1275, 1348, 1423, 1500, 1579, 1660, 1743, 1828, 1915, 2004, 2095, 2188

Offset: 0

N. J. A. Sloane, Mar 24 2007

Numbers m such that m + 21 is a square. The product of two consecutive terms belongs to the sequence, see formula. - Klaus Purath, Oct 30 2022

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

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

A row of A127080.

List([0..60], n-> (n-8)^2 -21); # G. C. Greubel, Aug 12 2019

[n^2-16*n+43: n in [0..60]]; // Vincenzo Librandi, Nov 12 2014

seq((n-8)^2 -21, n=0..60); # G. C. Greubel, Aug 12 2019

CoefficientList[Series[(60x^2 -101x +43)/(1-x)^3, {x,0,60}], x] (* Vincenzo Librandi, Nov 12 2014 *)
(Range[0,60] -8)^2 -21 (* G. C. Greubel, Aug 12 2019 *)

Vec(-(60*x^2-101*x+43)/(x-1)^3 + O(x^60)) \\ Colin Barker, Nov 12 2014

[(n-8)^2 -21 for n in (0..60)] # G. C. Greubel, Aug 12 2019

a(n) = n^2 - 16*n + 43.
From Colin Barker, Nov 12 2014: (Start)
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3).
G.f.: (43 -101*x + 60*x^2)/(1-x)^3. (End)
E.g.f.: (43 - 15*x + x^2)*exp(x). - G. C. Greubel, Aug 12 2019
From Klaus Purath, Oct 30 2022: (Start)
According to the formula a(n) = n^2 - 16*n + 43 when expanded to negative indices, a(n)*a(n+1) = a(a(n)+n) = (a(n)+n)*(a(n+1)-(n+1)) + 43.
a(n) = 2*a(n-1) - a(n-2) + 2. (End)

A127146 Q(n,4), where Q(m,k) is defined in A127080 and A127137.

Original entry on oeis.org

12, 3, -4, -9, -12, -13, -12, -9, -4, 3, 12, 23, 36, 51, 68, 87, 108, 131, 156, 183, 212, 243, 276, 311, 348, 387, 428, 471, 516, 563, 612, 663, 716, 771, 828, 887, 948, 1011, 1076, 1143, 1212, 1283, 1356, 1431, 1508, 1587, 1668, 1751, 1836, 1923, 2012, 2103, 2196, 2291

Offset: 0

N. J. A. Sloane, Mar 24 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 (3,-3,1).

A row of A127080.

List([0..60], n-> (n-5)^2 -13); # G. C. Greubel, Aug 25 2019

[(n-5)^2 -13: n in [0..60]]; // G. C. Greubel, Aug 25 2019

seq((n-5)^2 -13, n=0..60); # G. C. Greubel, Aug 25 2019

Table[n^2-10n+12,{n,0,60}]  (* Harvey P. Dale, Apr 02 2011 *)
Range[-5, 55]^2 - 13 (* G. C. Greubel, Aug 25 2019 *)

a(n)=n^2-10*n+12 \\ Charles R Greathouse IV, Jun 17 2017

[lucas_number2(2,n,6-n) for n in range(-6,48)] # Zerinvary Lajos, Mar 12 2009

a(n) = n^2 - 10*n + 12.
a(n) = a(n-1) + 2*n - 11, with a(0)=12. - Vincenzo Librandi, Nov 23 2010
G.f.: (12 - 33*x + 23*x^2)/(1 - x)^3. - Harvey P. Dale, Apr 02 2011
E.g.f.: (12 - 9*x + x^2)*exp(x). - G. C. Greubel, Aug 25 2019

A127148 Q(n,6), where Q(m,k) is defined in A127080 and A127137.

Original entry on oeis.org

-120, -15, 48, 75, 72, 45, 0, -57, -120, -183, -240, -285, -312, -315, -288, -225, -120, 33, 240, 507, 840, 1245, 1728, 2295, 2952, 3705, 4560, 5523, 6600, 7797, 9120, 10575, 12168, 13905, 15792, 17835, 20040, 22413, 24960, 27687, 30600, 33705, 37008, 40515, 44232, 48165

Offset: 0

N. J. A. Sloane, Mar 24 2007

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

Colin Barker, 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 A127080.

List([0..50], n-> n^3 -24*n^2 +128*n -120); # G. C. Greubel, Aug 12 2019

[n^3 -24*n^2 +128*n -120: n in [0..50]]; // G. C. Greubel, Aug 12 2019

seq(n^3 -24*n^2 +128*n -120, n=0..50); # G. C. Greubel, Aug 12 2019

Table[n^3-24n^2+128n-120,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{-120,-15,48,75},50] (* Harvey P. Dale, Oct 22 2013 *)

Vec(3*(91*x^3-204*x^2+155*x-40)/(x-1)^4 + O(x^50)) \\ Colin Barker, Nov 11 2014

[n^3 -24*n^2 +128*n -120 for n in (0..50)] # G. C. Greubel, Aug 12 2019

a(n) = n^3 -24*n^2 +128*n -120.
a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4), a(0)=-120, a(1)=-15, a(2)=48, a(3)=75. - Harvey P. Dale, Oct 22 2013
G.f.: (-3)*(40-155*x+204*x^2-91*x^3)/(1-x)^4. - Colin Barker, Nov 11 2014
E.g.f.: (-120 + 105*x - 21*x^2 + x^3)*exp(x). - G. C. Greubel, Aug 12 2019

A105937 Infinite square array read by antidiagonals: T(m, 0) = 1, T(m, 1) = m; T(m, k) = (m - k + 1) T(m+1, k-1) - (k-1) (m+1) T(m+2, k-2).

Original entry on oeis.org

1, 1, 0, 1, 1, -2, 1, 2, -2, 0, 1, 3, 0, -12, 36, 1, 4, 4, -24, 24, 0, 1, 5, 10, -30, -60, 420, -1800, 1, 6, 18, -24, -216, 720, -720, 0, 1, 7, 28, 0, -420, 420, 5040, -30240, 176400, 1, 8, 40, 48, -624, -960, 14400, -40320, 40320, 0, 1, 9, 54, 126, -756, -3780, 22680, 22680, -589680, 3764880, -28576800

Offset: 0

Vincent v.d. Noort, Mar 24 2007

			Array begins
   1  1  1   1   1   1   1   1   1   1 ... (A000012)
   0  1  2   3   4   5   6   7   8   9 ... (A001477)
  -2 -2  0   4  10  18  28  40  54  70 ... (A028552)
   0 12 24  30  24   0  48 126 240 396 ... (A126935)
  36 24 60 216 420 624 756 720 396 360 ... (A126958)
...

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

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

Rows give A000027, A028552, A126935, A126958.
Columns give A126934, A126962, A127067, A127068, A127070.
A127080 gives another version of the array.

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(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 28 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(seq(T(n-k, k), k=0..n), n=0..12); # G. C. Greubel, Jan 28 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-k,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 28 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) )); \\ G. C. Greubel, Jan 28 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(n-k, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 28 2020

See A127080 for e.g.f.

More terms added by G. C. Greubel, Jan 28 2020

A127137 Define an array by 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). Sequence gives Q(0,n).

Original entry on oeis.org

1, 1, -2, -5, 12, 43, -120, -531, 1680, 8601, -30240, -172965, 665280, 4161555, -17297280, -116658675, 518918400, 3735104625, -17643225600, -134498225925, 670442572800, 5380583766075, -28158588057600, -236759435017875, 1295295050649600, 11364769115001225, -64764752532480000

Offset: 0

N. J. A. Sloane, Mar 24 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..500

A001813 interleaved with A090470.

Column 0 of array A127080.

function b(n)
  if n mod 2 eq 0 then return Factorial(n)/Gamma(n/2+1);
  else return 2^(n-1)*Gamma((n+1)/2)*(&+[Binomial(2*j,j)/8^j: j in [0..((n-1)/2)]]);
  end if; return b; end function;
[Round((-1)^Binomial(n, 2)*b(n)): n in [0..30]]; // G. C. Greubel, Jan 30 2020

seq( (-1)^binomial(n,2)*(`if`(`mod`(n,2)=0, n!/(n/2)!, 2^(n-1)*((n-1)/2)!*add( binomial(2*j,j)/8^j, j=0..((n-1)/2)) ) ), n=0..30); # G. C. Greubel, Jan 30 2020

Q[0, k_]:= (-1)^Binomial[k, 2]*If[EvenQ[k], k!/(k/2)!, 2^((k-1)/2)*(k)!! Beta[1/2, 1/2, (k+1)/2]/Sqrt[2]]//FullSimplify; Table[Q[0, k], {k, 0, 30}] (* G. C. Greubel, Jan 30 2020 *)

a(n) = (-1)^binomial(n, 2)*if(n%2==0, n!/(n/2)!, 2^(n-1)*((n-1)/2)!*sum( j=0, (n-1)/2, binomial(2*j,j)/8^j));
vector(31, n, a(n-1)) \\ G. C. Greubel, Jan 30 2020

@CachedFunction
def b(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 a(k): return (-1)^binomial(k, 2)*b(k)
[a(n) for n in (0..30)] # G. C. Greubel, Jan 30 2020

See A127080 for e.g.f..

a(n) = (-1)^binomial(n,2)*b(n), where b(2*n) = (2*n)!/n! and b(2*n+1) = 4^n*n!* Sum_{j=0..n} binomial(2*j,j)/8^j. - G. C. Greubel, Jan 30 2020

Typo in name corrected 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.

cp's OEIS Frontend

A127138 Q(1,n), where Q(m,k) is defined in A127080 and A127137.

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A127144 Q(2,n), where Q(m,k) is defined in A127080 and A127137.

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A127145 Q(3,n), where Q(m,k) is defined in A127080 and A127137.

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A127147 Q(n,5), where Q(m,k) is defined in A127080 and A127137.

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A127146 Q(n,4), where Q(m,k) is defined in A127080 and A127137.

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A127148 Q(n,6), where Q(m,k) is defined in A127080 and A127137.

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A127137 Define an array by 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). Sequence gives Q(0,n).