A047851 -id:A047851 - cp's OEIS frontend

A047848 Array A read by diagonals; n-th difference of (A(k,n), A(k,n-1),..., A(k,0)) is (k+2)^(n-1), for n=1,2,3,...; k=0,1,2,...

1, 2, 1, 5, 2, 1, 14, 6, 2, 1, 41, 22, 7, 2, 1, 122, 86, 32, 8, 2, 1, 365, 342, 157, 44, 9, 2, 1, 1094, 1366, 782, 260, 58, 10, 2, 1, 3281, 5462, 3907, 1556, 401, 74, 11, 2, 1, 9842, 21846, 19532, 9332, 2802, 586, 92, 12, 2, 1, 29525, 87382, 97657, 55988, 19609, 4682, 821, 112, 13, 2, 1

Offset: 0

Clark Kimberling

			Array, A(n, k), begins as:
  1, 2,  5,  14,   41, ... = A007051.
  1, 2,  6,  22,   86, ... = A047849.
  1, 2,  7,  32,  157, ... = A047850.
  1, 2,  8,  44,  260, ... = A047851.
  1, 2,  9,  58,  401, ... = A047852.
  1, 2, 10,  74,  586, ... = A047853.
  1, 2, 11,  92,  821, ... = A047854.
  1, 2, 12, 112, 1112, ... = A047855.
  1, 2, 13, 134, 1465, ... = A047856.
  1, 2, 14, 158, 1886, ... = A196791.
  1, 2, 15, 184, 2381, ... = A196792.
Downward antidiagonals, T(n, k), begins as:
      1;
      2,     1;
      5,     2,     1;
     14,     6,     2,     1;
     41,    22,     7,     2,     1;
    122,    86,    32,     8,     2,    1;
    365,   342,   157,    44,     9,    2,   1;
   1094,  1366,   782,   260,    58,   10,   2,   1;
   3281,  5462,  3907,  1556,   401,   74,  11,   2,  1;
   9842, 21846, 19532,  9332,  2802,  586,  92,  12,  2, 1;
  29525, 87382, 97657, 55988, 19609, 4682, 821, 112, 13, 2, 1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A047857 (row sums), A196793 (main diagonal).

A:= func< n,k | ((n+3)^k +n+1)/(n+2) >; // array A047848
[A(k,n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 11 2025

A[n_, k_]:= ((n+3)^k +n+1)/(n+2);
A047848[n_, k_]:= A[k,n-k];
Table[A047848[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 11 2025 *)

def A(n,k): return (pow(n+3,k) +n+1)//(n+2) # array A047848
print(flatten([[A(k,n-k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 11 2025

A(n, k) = ((n+3)^k + n + 1)/(n+2). - Ralf Stephan, Feb 14 2004
From G. C. Greubel, Jan 11 2025: (Start)
T(n, k) = ((k+3)^(n-k) + k + 1)/(k+2) (antidiagonal triangle).
T(n, n) = A196793(n).
Sum_{k=0..n} T(n, k) = A047857(n). (End)

A123490 Triangle whose k-th column satisfies a(n) = (k+3)a(n-1)-(k+2)a(n-2).

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 5, 2, 1, 16, 14, 6, 2, 1, 32, 41, 22, 7, 2, 1, 64, 122, 86, 32, 8, 2, 1, 128, 365, 342, 157, 44, 9, 2, 1, 256, 1094, 1366, 782, 260, 58, 10, 2, 1, 512, 3281, 5462, 3907, 1556, 401, 74, 11, 2, 1, 1024, 9842, 21846, 19532, 9332, 2802, 586, 92, 12, 2, 1

Offset: 0

Paul Barry, Oct 01 2006

			Triangle begins
     1;
     2,    1;
     4,    2,     1;
     8,    5,     2,     1;
    16,   14,     6,     2,    1;
    32,   41,    22,     7,    2,    1;
    64,  122,    86,    32,    8,    2,   1;
   128,  365,   342,   157,   44,    9,   2,  1;
   256, 1094,  1366,   782,  260,   58,  10,  2,  1;
   512, 3281,  5462,  3907, 1556,  401,  74, 11,  2, 1;
  1024, 9842, 21846, 19532, 9332, 2802, 586, 92, 12, 2, 1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Columns include A000079, A007051, A047849, A047850, A047851.

Cf. A047848, A103439 (row sums), A123491 (diagonal sums).

[((k+2)^(n-k) +k)/(k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 15 2021

Table[((k+2)^(n-k) +k)/(k+1), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 14 2017 *)

for(n=0, 10, for(k=0,n, print1(((k+2)^(n-k)+k)/(k+1), ", "))) \\ G. C. Greubel, Oct 14 2017

flatten([[((k+2)^(n-k) +k)/(k+1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 15 2021

Column k has g.f.: x^k*(1-x(1+k))/((1-x)*(1-x(2+k))).
T(n,k) = ((k+2)^(n-k) + k)/(k+1), for 0 <= k <= n.
Sum_{k=0..n} T(n, k) = A103439(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A123491(n).

A196791 a(n) = A047848(9, n).

Original entry on oeis.org

1, 2, 14, 158, 1886, 22622, 271454, 3257438, 39089246, 469070942, 5628851294, 67546215518, 810554586206, 9726655034462, 116719860413534, 1400638324962398, 16807659899548766, 201691918794585182, 2420303025535022174, 29043636306420266078, 348523635677043192926

Offset: 0

Vincenzo Librandi, Oct 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (13,-12).

Cf. A047848, A047849, A047850, A047851, A047852, A047853, A047854, A047855, A047856.

Cf. A001021 (first differences).

Magma
```
[(12^n+10)/11: n in [0..20]];
```

LinearRecurrence[{13,-12},{1,2},30] (* Harvey P. Dale, Sep 07 2015 *)
(12^Range[0,40] +10)/11 (* G. C. Greubel, Jan 17 2025 *)

def A196791(n): return (pow(12, n) + 10)//11
print([A196791(n) for n in range(41)]) # G. C. Greubel, Jan 17 2025

a(n) = (12^n + 10)/11.
a(n) = 12*a(n-1) - 10, with a(0) = 1.
G.f.: (1-11*x)/((1-x)*(1-12*x)). - Bruno Berselli, Oct 11 2011
From Elmo R. Oliveira, Aug 30 2024: (Start)
E.g.f.: exp(x)*(exp(11*x) + 10)/11.
a(n) = 13*a(n-1) - 12*a(n-2) for n > 1. (End)

A196792 a(n) = A047848(10, n).

Original entry on oeis.org

1, 2, 15, 184, 2381, 30942, 402235, 5229044, 67977561, 883708282, 11488207655, 149346699504, 1941507093541, 25239592216022, 328114698808275, 4265491084507564, 55451384098598321, 720867993281778162, 9371283912663116095, 121826690864620509224, 1583746981240066619901

Offset: 0

Vincenzo Librandi, Oct 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (14,-13).

Cf. A047848, A047849, A047850, A047851, A047852, A047853, A047854, A047855, A047856.

Cf. A001022 (first differences).

Magma
```
[(13^n+11)/12: n in [0..20]];
```

(13^Range[0,40] +11)/12 (* G. C. Greubel, Jan 17 2025 *)

def A196792(n): return (pow(13, n) + 11)//12
print([A196792(n) for n in range(41)]) # G. C. Greubel, Jan 17 2025

a(n) = (13^n + 11)/12.
a(n) = 13*a(n-1) - 11, with a(0) = 1.
G.f.: (1-12*x)/((1-x)*(1-13*x)). - Bruno Berselli, Oct 11 2011
From Elmo R. Oliveira, Aug 30 2024: (Start)
E.g.f.: exp(x)*(exp(12*x) + 11)/12.
a(n) = 14*a(n-1) - 13*a(n-2) for n > 1. (End)

A196793 a(n) = A047848(n, n).

Original entry on oeis.org

1, 2, 7, 44, 401, 4682, 66431, 1111112, 21435889, 469070942, 11488207655, 311505013052, 9267595563617, 300239975158034, 10523614159962559, 396861212733968144, 16024522975978953761, 689852631578947368422, 31544039619835776489479

Offset: 0

Vincenzo Librandi, Oct 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A047848, A047849, A047850, A047851, A047852, A047853, A047854, A047855, A047856

Magma
```
[((n+3)^n+n+1)/(n+2): n in [0..20]]
```

A196793:=n->((n+3)^n+n+1)/(n+2); seq(A196793(k), k=0..50); # Wesley Ivan Hurt, Nov 08 2013

Table[((n+3)^n+n+1)/(n+2), {n,0,50}] (* Wesley Ivan Hurt, Nov 08 2013 *)

def A196793(n): return (pow(n+3, n) +n+1)//(n+2)
print([A196793(n) for n in range(51)]) # G. C. Greubel, Jan 17 2025

a(n) = ((n+3)^n + n + 1)/(n+2).

A142596 Triangle T(n, k) = T(n-1, k-1) + 3T(n-1, k) + 2T(n-1, k-1), with T(n,1) = T(n, n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 21, 21, 1, 1, 66, 126, 66, 1, 1, 201, 576, 576, 201, 1, 1, 606, 2331, 3456, 2331, 606, 1, 1, 1821, 8811, 17361, 17361, 8811, 1821, 1, 1, 5466, 31896, 78516, 104166, 78516, 31896, 5466, 1, 1, 16401, 112086, 331236, 548046, 548046, 331236, 112086, 16401, 1

Offset: 1

Roger L. Bagula, Sep 22 2008

			The triangle begins as:
  1;
  1,     1;
  1,     6,      1;
  1,    21,     21,      1;
  1,    66,    126,     66,      1;
  1,   201,    576,    576,    201,      1;
  1,   606,   2331,   3456,   2331,    606,      1;
  1,  1821,   8811,  17361,  17361,   8811,   1821,      1;
  1,  5466,  31896,  78516, 104166,  78516,  31896,   5466,     1;
  1, 16401, 112086, 331236, 548046, 548046, 331236, 112086, 16401, 1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A008292, A047851, A060187, A119258.

function T(n,k)
  if k eq 1 or k eq n then return 1;
  else return T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k-1);
  end if; return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 13 2021

T[n_, k_]:= T[n,k]= If[k==1 || k==n, 1, T[n-1, k-1] +3*T[n-1, k] +2*T[n-1, k-1]];
Table[T[n, k], {n, 10}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 13 2021 *)

@CachedFunction
def T(n,k): return 1 if k==1 or k==n else T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k-1)
flatten([[T(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 13 2021

T(n, k) = T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k-1), with T(n,1) = T(n, n) = 1.

Sum_{k=1..n} T(n, k) = (6^(n-1) + 4)/5 = A047851(n-1). - G. C. Greubel, Apr 13 2021

Edited by G. C. Greubel, Apr 13 2021

A166065 Triangle, read by rows, given by [0,1,1,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 2, 0, 2, 2, 0, 4, 2, 2, 0, 8, 4, 2, 2, 0, 16, 8, 4, 2, 2, 0, 32, 16, 8, 4, 2, 2, 0, 64, 32, 16, 8, 4, 2, 2, 0, 128, 64, 32, 16, 8, 4, 2, 2, 0, 256, 128, 64, 32, 16, 8, 4, 2, 2, 0, 512, 256, 128, 64, 32, 16, 8, 4, 2, 2, 0, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 2, 0, 2048, 1024

Offset: 0

Philippe Deléham, Oct 05 2009

			Triangle begins :
1,
0,2,
0,2,2,
0,4,2,2,
0,8,4,2,2,
0,16,8,4,2,2,
0,32,16,8,4,2,2,
0,64,32,16,8,4,2,2,
0,128,64,32,16,8,4,2,2,
0,256,128,64,32,16,8,4,2,2,
0,512,256,128,64,32,16,8,4,2,2,

Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A084247(n), A000007(n), A000079(n), A001787(n+1), A166060(n), A165665(n), A083585(n) for x= -1, 0, 1, 2, 3, 4, 5 respectively. Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A040000(n), A000079(n), A095121(n), A047851(n), A047853(n), A047855(n) for x = 0, 1, 2, 3, 4, 5 respectively.

G.f.: (1-2*x+x*y)/((-1+2*x)*(x*y-1)). - R. J. Mathar, Aug 11 2015

A166124 Triangle, read by rows, given by [0,1/2,1/2,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 2, 0, 1, 2, 0, 1, 1, 2, 0, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 0

Philippe Deléham, Oct 07 2009

			Triangle begins :
1 ;
0,2 ;
0,1,2 ;
0,1,1,2 ;
0,1,1,1,2 ;
0,1,1,1,1,2 ;
0,1,1,1,1,1,2 ; ...

Sum_{k, 0<=k<=n} T(n,k)*x^(n-k)= A166122(n), A166114(n), A084222(n), A084247(n), A000034(n), A040000(n), A000027(n+1), A000079(n), A007051(n), A047849(n), A047850(n), A047851(n), A047852(n), A047853(n), A047854(n), A047855(n), A047856(n) for x= -5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^k= A000007(n), A000027(n+1), A033484(n), A134931(n), A083597(n) for x= 0,1,2,3,4 respectively.
T(n,k)= A166065(n,k)/2^(n-k).
G.f.: (1-x+x*y)/(1-x-x*y+x^2*y). - Philippe Deléham, Nov 09 2013
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 09 2013

cp's OEIS Frontend

A047848 Array A read by diagonals; n-th difference of (A(k,n), A(k,n-1),..., A(k,0)) is (k+2)^(n-1), for n=1,2,3,...; k=0,1,2,...

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A123490 Triangle whose k-th column satisfies a(n) = (k+3)*a(n-1)-(k+2)*a(n-2).

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A196791 a(n) = A047848(9, n).

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A196792 a(n) = A047848(10, n).

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A196793 a(n) = A047848(n, n).

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A142596 Triangle T(n, k) = T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k-1), with T(n,1) = T(n, n) = 1, read by rows.

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A166065 Triangle, read by rows, given by [0,1,1,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A166124 Triangle, read by rows, given by [0,1/2,1/2,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

A123490 Triangle whose k-th column satisfies a(n) = (k+3)a(n-1)-(k+2)a(n-2).

A142596 Triangle T(n, k) = T(n-1, k-1) + 3T(n-1, k) + 2T(n-1, k-1), with T(n,1) = T(n, n) = 1, read by rows.