A127137 -id:A127137 - 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

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).

Original entry on oeis.org

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

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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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).