cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A134060 Triangle T(n,k) = A124927(n,k) + A134058(n,k) - A007318(n,k), read by rows.

Original entry on oeis.org

1, 2, 3, 2, 6, 3, 2, 9, 9, 3, 2, 12, 18, 12, 3, 2, 15, 30, 30, 15, 3, 2, 18, 45, 60, 45, 18, 3, 2, 21, 63, 105, 105, 63, 21, 3, 2, 24, 84, 168, 210, 168, 84, 24, 3, 2, 27, 108, 252, 378, 378, 252, 108, 27, 3
Offset: 0

Views

Author

Gary W. Adamson, Oct 05 2007

Keywords

Examples

			First few rows of the triangle are:
  1;
  2,  3;
  2,  6,  3;
  2,  9,  9,  3;
  2, 12, 18, 12,  3;
  2, 15, 30, 30, 15, 3;
  ...

Links

Crossrefs

Cf. A007318, A052940 (row sums), A127927, A134058.

Programs

  • Magma
    [1] cat [k eq 0 select 2 else 3*Binomial(n,k): k in [0..n], n in [1..12]]; // G. C. Greubel, May 03 2021
  • Mathematica
    Table[3*Binomial[n, k] -Boole[k==0] -Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 03 2021 *)
  • Sage
    def A134060(n,k): return 3*binomial(n,k) -bool(k==0) -bool(n==0)
flatten([[A134060(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 03 2021

Formula

T(n,k) = A124927(n,k) + A134058(n,k) - A007318(n,k) as infinite lower triangular matrices.
Sum_{k=0..n} T(n, k) = A052940(n).
T(n, k) = 3*binomial(n,k) - [k=0] - [n=0]. - G. C. Greubel, May 03 2021

A210042 Triangle of coefficients of polynomials u(n,x) jointly generated with A124927; see the Formula section.

Original entry on oeis.org

1, 3, 5, 2, 7, 6, 2, 9, 12, 8, 2, 11, 20, 20, 10, 2, 13, 30, 40, 30, 12, 2, 15, 42, 70, 70, 42, 14, 2, 17, 56, 112, 140, 112, 56, 16, 2, 19, 72, 168, 252, 252, 168, 72, 18, 2, 21, 90, 240, 420, 504, 420, 240, 90, 20, 2, 23, 110, 330, 660, 924, 924, 660, 330, 110
Offset: 1

Views

Author

Clark Kimberling, Mar 17 2012

Keywords

Comments

Row sums: A000225
For a discussion and guide to related arrays, see A208510.
u(n,x) = u(n-1,x) + v(n-1,x) + 1,
v(n,x) = x*u(n-1,x) + x*v(n-1,x) + 1,
where u(1,x)=1, v(1,x)=1.
Subtriangle of the triangle given by (1, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012

Examples

			From _Philippe Deléham_, Mar 25 2012: (Start)
(1, 2, -2, 1, 0, 0, ...) DELTA (0, 0, 1, 0, 0, ...) begins:
   1;
   1,  0;
   3,  0,  0;
   5,  2,  0,  0;
   7,  6,  2,  0,  0;
   9, 12,  8,  2,  0,  0;
  11, 20, 20, 10,  2,  0,  0;
  13, 30, 40, 30, 12,  2,  0,  0; (End)

Crossrefs

Cf. A124927, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210042 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A124927 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A010701 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A000012 signed *)

Formula

First five rows:
1;
3,
5, 2;
7, 6, 2;
9, 12, 8, 2;
First three polynomials u(n,x): 1, 3, 5 + 2x.
Also, counting the top row as row 0, row n for n > 0 is as follows: 2n+1, 2C(n,2), 2C(n,3), ..., 2C(n,n).
From Philippe Deléham, Mar 25 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1-x-y*x+2*x^2)/(1-2*x-y*x+x^2+y*x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,0) = 1, T(2,0) = 3, T(1,1) = T(2,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
G.f.: (1+x-x*y)*x*y/((-1+x)*(x+x*y-1)). - R. J. Mathar, Aug 12 2015

A134065 Triangle read by rows: A134059 + A124927 - A007318 as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 4, 3, 8, 4, 3, 12, 12, 4, 3, 16, 24, 16, 4, 3, 20, 40, 40, 20, 4, 3, 24, 60, 80, 60, 24, 4, 3, 28, 84, 140, 140, 84, 28, 4, 3, 32, 112, 224, 280, 224, 112, 32, 4, 3, 36, 144, 336, 504, 504, 336, 144, 36, 4, 3, 40, 180, 480, 840, 1008, 840, 480, 180, 40, 4, 3, 44, 220, 660, 1320, 1848, 1848, 1320, 660, 220, 44, 4
Offset: 0

Views

Author

Gary W. Adamson, Oct 05 2007

Keywords

Comments

Row sums = A078485 starting (1, 7, 15, 31, 63, 127, ...).

Examples

			First few rows of the triangle:
  1;
  3,  4;
  3,  8,  4;
  3, 12, 12,  4;
  3, 16, 24, 16,  4;
  3, 20, 40, 40, 20,  4;
  ...

Crossrefs

Cf. A078485, A124927, A134059.

Extensions

a(15) = 3 inserted and more terms from Georg Fischer, Jun 07 2023

A208510 Triangle of coefficients of polynomials u(n,x) jointly generated with A029653; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 7, 9, 5, 1, 1, 9, 16, 14, 6, 1, 1, 11, 25, 30, 20, 7, 1, 1, 13, 36, 55, 50, 27, 8, 1, 1, 15, 49, 91, 105, 77, 35, 9, 1, 1, 17, 64, 140, 196, 182, 112, 44, 10, 1, 1, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1, 1, 21, 100, 285, 540, 714, 672, 450, 210, 65, 12, 1
Offset: 1

Views

Author

Clark Kimberling, Feb 28 2012

Keywords

Comments

Row sums: A083329
Alternating row sums: 1,0,-1,-1,-1,-1,-1,-1,-1,-1,...
Antidiagonal sums: A000071 (-1+Fibonacci numbers)
col 1: A000012
col 2: A005408
col 3: A000290
col 4: A000330
col 5: A002415
col 6: A005585
col 7: A040977
col 8: A050486
col 9: A053347
col 10: A054333
col 11: A054334
col 12: A057788
col 2n-1 of A208510 is column n of A208508
col 2n of A208510 is column n of A208509.
...
GENERAL DISCUSSION:
A208510 typifies arrays generated by paired recurrence equations of the following form:
u(n,x)=a(n,x)*u(n-1,x)+b(n,x)*v(n-1,x)+c(n,x)
v(n,x)=d(n,x)*u(n-1,x)+e(n,x)*v(n-1,x)+f(n,x).
...
These first-order recurrences imply separate second-order recurrences. In order to show them, the six functions a(n,x),...,f(n,x) are abbreviated as a,b,c,d,e,f.
Then, starting with initial values u(1,x)=1 and u(2,x)=a+b+c: u(n,x) = (a+e)u(n-1,x) + (bd-ae)u(n-2,x) + bf-ce+c.
With initial values v(1,x)=1 and v(2,x)=d+e+f: v(n,x) = (a+e)v(n-1,x) + (bd-ae)v(n-2,x) + cd-af+f.
...
In the guide below, the last column codes certain sequences that occur in one of these ways: row, column, edge, row sum, alternating row sum. Coding:
A: 1,-1,1,-1,1,-1,1.... A033999
B: 1,2,4,8,16,32,64,... powers of 2
C: 1,1,1,1,1,1,1,1,.... A000012
D: 2,2,2,2,2,2,2,2,.... A007395
E: 2,4,6,8,10,12,14,... even numbers
F: 1,1,2,3,5,8,13,21,.. Fibonacci numbers
N: 1,2,3,4,5,6,7,8,.... A000027
O: 1,3,5,7,9,11,13,.... odd numbers
P: 1,3,9,27,81,243,.... powers of 3
S: 1,4,9,16,25,36,49,.. squares
T: 1,3,6,10,15,21,38,.. triangular numbers
Z: 1,0,0,0,0,0,0,0,0,.. A000007
*: (eventually) periodic alternating row sums
^: has a limiting row; i.e., the polynomials "approach" a power series
This coding includes indirect and repeated occurrences; e.g. F occurs thrice at A094441: in column 1 directly as Fibonacci numbers, in row sums as odd-indexed Fibonacci numbers, and in alternating row sums as signed Fibonacci numbers.
......... a....b....c....d....e....f....code
A034839 u 1....1....0....1....x....0....CCOT
A034867 v 1....1....0....1....x....0....CEN
A210221 u 1....1....0....1....2x...0....BBFF
A210596 v 1....1....0....1....2x...0....BBFF
A105070 v 1....2x...0....1....1....0....BN
A207605 u 1....1....0....1....x+1..0....BCFFN
A106195 v 1....1....0....1....x+1..0....BCFFN
A207606 u 1....1....0....x....x+1..0....DNT
A207607 v 1....1....0....x....x+1..0....DNT
A207608 u 1....1....0....2x...x+1..0....N
A207609 v 1....1....0....2x...x+1..0....C
A207610 u 1....1....0....1....x....1....CF
A207611 v 1....1....0....1....x....1....BCF
A207612 u 1....1....0....1....2x...1....BF
A207613 v 1....1....0....1....2x...1....BF
A207614 u 1....1....0....1....x+1..1....CN
A207615 v 1....1....0....1....x+1..1....CFN
A207616 u 1....1....0....x....1....1....CE
A207617 v 1....1....0....x....1....1....CNO
A029638 u 1....1....0....x....x....1....CDNO
A029635 v 1....1....0....x....x....1....CDNOZ
A207618 u 1....1....0....x....2x...1....N
A207619 v 1....1....0....x....2x...1....CFN
A207620 u 1....1....0....x....x+1..1....DET
A207621 v 1....1....0....x....x+1..1....DNO
A207622 u 1....1....0....2x...1....1....BT
A207623 v 1....1....0....2x...1....1....BN
A207624 u 1....1....0....2x...x....1....N
A102662 v 1....1....0....2x...x....1....CO
A207625 u 1....1....0....2x...x+1..1....T
A207626 v 1....1....0....2x...x+1..1....N
A207627 u 1....1....0....2x...2x...1....BN
A207628 v 1....1....0....2x...2x...1....BCE
A207629 u 1....1....0....x+1..1....1....CET
A207630 v 1....1....0....x+1..1....1....CO
A207631 u 1....1....0....x+1..x....1....DF
A207632 v 1....1....0....x+1..x....1....DEF
A207633 u 1....1....0....x+1..2x...1....F
A207634 v 1....1....0....x+1..2x...1....F
A207635 u 1....1....0....x+1..x+1..1....DN
A207636 v 1....1....0....x+1..x+1..1....CD
A160232 u 1....x....0....1....2x...0....BCFN
A208341 v 1....x....0....1....2x...0....BCFFN
A085478 u 1....x....0....1....x+1..0....CCOFT*
A078812 v 1....x....0....1....x+1..0....CEFN*
A208342 u 1....x....0....x....x....0....CCFNO
A208343 v 1....x....0....x....x....0....BBCDFZ
A208344 u 1....x....0....x....2x...0....CCFN
A208345 v 1....x....0....x....2x...0....CFZ
A094436 u 1....x....0....x....x+1..0....CFFN
A094437 v 1....x....0....x....x+1..0....CEFF
A117919 u 1....x....0....2x...1....0....BCNT
A135837 v 1....x....0....2x...1....0....BCET
A208328 u 1....x....0....2x...x....0....CCOP
A208329 v 1....x....0....2x...x....0....DPZ
A208330 u 1....x....0....2x...x+1..0....CNPT
A208331 v 1....x....0....2x...x+1..0....CN
A208332 u 1....x....0....2x...2x...0....CCE
A208333 v 1....x....0....2x...2x...0....DZ
A208334 u 1....x....0....x+1..1....0....CCNT
A208335 v 1....x....0....x+1..1....0....CCN*
A208336 u 1....x....0....x+1..x....0....CFNT*
A208337 v 1....x....0....x+1..x....0....ACFN*
A208338 u 1....x....0....x+1..2x...0....CNP
A208339 v 1....x....0....x+1..2x...0....BCNP
A202390 u 1....x....0....x+1..x+1..0....CFPTZ*
A208340 v 1....x....0....x+1..x+1..0....FNPZ*
A208508 u 1....x....0....1....1....1....CCES
A208509 v 1....x....0....1....1....1....BCO
A208510 u 1....x....0....1....x....1....CCCNOS*
A029653 v 1....x....0....1....x....1....BCDOSZ*
A208511 u 1....x....0....1....2x...1....BCFO
A208512 v 1....x....0....1....2x...1....BDFO
A208513 u 1....x....0....1....x+1..1....CCES*
A111125 v 1....x....0....1....x+1..1....COO*
A133567 u 1....x....0....x....1....1....CCOTT
A133084 v 1....x....0....x....1....1....BBCEN
A208514 u 1....x....0....x....x....1....CEFN
A208515 v 1....x....0....x....x....1....BCDFN
A208516 u 1....x....0....x....2x...1....CNN
A208517 v 1....x....0....x....2x...1....CCN
A208518 u 1....x....0....x....x+1..1....CFNT
A208519 v 1....x....0....x....x+1..1....NFFT
A208520 u 1....x....0....2x...1....1....BCTT
A208521 v 1....x....0....2x...1....1....BEN
A208522 u 1....x....0....2x...x....1....CCN
A208523 v 1....x....0....2x...x....1....CCO
A208524 u 1....x....0....2x...x+1..1....CT*
A208525 v 1....x....0....2x...x+1..1....ACNP*
A208526 u 1....x....0....2x...2x...1....CEN
A208527 v 1....x....0....2x...2x...1....CCE
A208606 u 1....x....0....x+1..1....1....CCS
A208607 v 1....x....0....x+1..1....1....CNO
A208608 u 1....x....0....x+1..x....1....CFOT
A208609 v 1....x....0....x+1..x....1....DEN*
A208610 u 1....x....0....x+1..2x...1....CO
A208611 v 1....x....0....x+1..2x...1....DE
A208612 u 1....x....0....x+1..x+1..1....CFNS
A208613 v 1....x....0....x+1..x+1..1....CFN*
A105070 u 1....2x...0....1....1....0....BN
A207536 u 1....2x...0....1....1....0....BCT
A208751 u 1....2x...0....1....x+1..0....CDPT
A208752 v 1....2x...0....1....x+1..0....CNP
A135837 u 1....2x...0....x....1....0....BCNT
A117919 v 1....2x...0....x....1....0....BCNT
A208755 u 1....2x...0....x....x....0....BCDEP
A208756 v 1....2x...0....x....x....0....BCCOZ
A208757 u 1....2x...0....x....2x...0....CDEP
A208758 v 1....2x...0....x....2x...0....CCEPZ
A208763 u 1....2x...0....2x...x....0....CDOP
A208764 v 1....2x...0....2x...x....0....CCCP
A208765 u 1....2x...0....2x...x+1..0....CE
A208766 v 1....2x...0....2x...x+1..0....CC
A208747 u 1....2x...0....2x...2x...0....CDE
A208748 v 1....2x...0....2x...2x...0....CCZ
A208749 u 1....2x...0....x+1..1....0....BCOPT
A208750 v 1....2x...0....x+1..1....0....BCNP*
A208759 u 1....2x...0....x+1..2x....0...CE
A208760 v 1....2x...0....x+1..2x....0...BCO
A208761 u 1....2x...0....x+1..x+1...0...BCCT*
A208762 v 1....2x...0....x+1..x+1...0...BNZ*
A208753 u 1....2x...0....1....1.....1...BCS
A208754 v 1....2x...0....1....1.....1...BO
A105045 u 1....2x...0....1....2x....1...BCCOS*
A208659 v 1....2x...0....1....2x....1...BDOSZ*
A208660 u 1....2x...0....1....x+1...1...CDS
A208904 v 1....2x...0....1....x+1...1...CNO
A208905 u 1....2x...0....x....1.....1...BCT
A208906 v 1....2x...0....x....1.....1...BNN
A208907 u 1....2x...0....x....x.....1...BCN
A208756 v 1....2x...0....x....x.....1...BCCE
A208755 u 1....2x...0....x....2x....1...CEN
A208910 v 1....2x...0....x....2x....1...CCE
A208911 u 1....2x...0....x....x+1...1...BCT
A208912 v 1....2x...0....x....x+1...1...BNT
A208913 u 1....2x...0....2x...1.....1...BCT
A208914 v 1....2x...0....2x...1.....1...BEN
A208915 u 1....2x...0....2x...x.....1...CE
A208916 v 1....2x...0....2x...x.....1...CCO
A208919 u 1....2x...0....2x...x+1...1...CT
A208920 v 1....2x...0....2x...x+1...1...N
A208917 u 1....2x...0....2x...2x....1...CEN
A208918 v 1....2x...0....2x...2x....1...CCNP
A208921 u 1....2x...0....x+1..1.....1...BC
A208922 v 1....2x...0....x+1..1.....1...BON
A208923 u 1....2x...0....x+1..x.....1...BCNO
A208908 v 1....2x...0....x+1..x.....1...BDN*
A208909 u 1....2x...0....x+1..2x....1...BN
A208930 v 1....2x...0....x+1..2x....1...DN
A208931 u 1....2x...0....x+1..x+1...1...BCOS
A208932 v 1....2x...0....x+1..x+1...1...BCO*
A207537 u 1....x+1..0....1....1.....0...BCO
A207538 v 1....x+1..0....1....1.....0...BCE
A122075 u 1....x+1..0....1....x.....0...CCFN*
A037027 v 1....x+1..0....1....x.....0...CCFN*
A209125 u 1....x+1..0....1....2x....0...BCFN*
A164975 v 1....x+1..0....1....2x....0...BF
A209126 u 1....x+1..0....x....x.....0...CDFO*
A209127 v 1....x+1..0....x....x.....0...DFOZ*
A209128 u 1....x+1..0....x....2x....0...CDE*
A209129 v 1....x+1..0....x....2x....0...DEZ
A102756 u 1....x+1..0....x....x+1...0...CFNP*
A209130 v 1....x+1..0....x....x+1...0...CCFNP*
A209131 u 1....x+1..0....2x...x.....0...CDEP*
A209132 v 1....x+1..0....2x...x.....0...CNPZ*
A209133 u 1....x+1..0....2x...2x....0...CDN
A209134 v 1....x+1..0....2x...2x....0...CCN*
A209135 u 1....x+1..0....2x...x+1...0...CN*
A209136 v 1....x+1..0....2x...x+1...0...CCS*
A209137 u 1....x+1..0....x+1..x.....0...CFFP*
A209138 v 1....x+1..0....x+1..x.....0...AFFP*
A209139 u 1....x+1..0....x+1..2x....0...CF*
A209140 v 1....x+1..0....x+1..2x....0...BF
A209141 u 1....x+1..0....x+1..x+1...0...BCF*
A209142 v 1....x+1..0....x+1..x+1...0...BFZ*
A209143 u 1....x+1..0....1....1.....1...CCE*
A209144 v 1....x+1..0....1....1.....1...COO*
A209145 u 1....x+1..0....1....x.....1...CCFN*
A122075 v 1....x+1..0....1....x.....1...CCFN*
A209146 u 1....x+1..0....1....2x....1...BCF*
A209147 v 1....x+1..0....1....2x....1...BF
A209148 u 1....x+1..0....1....x+1...1...CCO*
A209149 v 1....x+1..0....1....x+1...1...CDO*
A209150 u 1....x+1..0....x....1.....1...CCNT*
A208335 v 1....x+1..0....x....1.....1...CDNN*
A209151 u 1....x+1..0....x....x.....1...CFN*
A208337 v 1....x+1..0....x....x.....1...ACFN*
A209152 u 1....x+1..0....x....2x....1...CN*
A208339 v 1....x+1..0....x....x.....1...BCN
A209153 u 1....x+1..0....x....x+1...1...CFT*
A208340 v 1....x+1..0....x....x.....1...FNZ*
A209154 u 1....x+1..0....2x...1.....1...BCT*
A209157 v 1....x+1..0....2x...1.....1...BNN
A209158 u 1....x+1..0....2x...x.....1...CN*
A209159 v 1....x+1..0....2x...x.....1...CO*
A209160 u 1....x+1..0....2x...2x....1...CN*
A209161 v 1....x+1..0....2x...2x....1...CE
A209162 u 1....x+1..0....2x...x+1...1...CT*
A209163 v 1....x+1..0....2x...x+1...1...CO*
A209164 u 1....x+1..0....x+1..1.....1...CC*
A209165 v 1....x+1..0....x+1..1.....1...CCN
A209166 u 1....x+1..0....x+1..x.....1...CFF*
A209167 v 1....x+1..0....x+1..x.....1...FF*
A209168 u 1....x+1..0....x+1..2x....1...CF*
A209169 v 1....x+1..0....x+1..2x....1...CF
A209170 u 1....x+1..0....x+1..x+1...1...CF*
A209171 v 1....x+1..0....x+1..x+1...1...CF*
A053538 u x....1....0....1....1.....0...BBCCFN
A076791 v x....1....0....1....1.....0...BBCDF
A209172 u x....1....0....1....2x....0...BCCFF
A209413 v x....1....0....1....2x....0...BCCFF
A094441 u x....1....0....1....x+1...0...CFFFN
A094442 v x....1....0....1....x+1...0...CEFFF
A054142 u x....1....0....x....x+1...0...CCFOT*
A172431 v x....1....0....x....x+1...0...CEFN*
A008288 u x....1....0....2x...1.....0...CCOO*
A035607 v x....1....0....2x...1.....0...ACDE*
A209414 u x....1....0....2x...x+1...0...CCS
A112351 v x....1....0....2x...x+1...0...CON
A209415 u x....1....0....x+1..x.....0...CCTN
A209416 v x....1....0....x+1..x.....0...ACN*
A209417 u x....1....0....x+1..2x....0...CC
A209418 v x....1....0....x+1..2x....0...BBC
A209419 u x....1....0....x+1..x+1...0...CFTZ*
A209420 v x....1....0....x+1..x+1...0...FNZ*
A209421 u x....1....0....1....1.....1...CCN
A209422 v x....1....0....1....1.....1...CD
A209555 u x....1....0....1....x.....1...CNN
A209556 v x....1....0....1....x.....1...CNN
A209557 u x....1....0....1....2x....1...BCN
A209558 v x....1....0....1....2x....1...BN
A209559 u x....1....0....1....x+1...1...CN
A209560 v x....1....0....1....x+1...1...CN
A209561 u x....1....0....x....1.....1...CCNNT*
A209562 v x....1....0....x....1.....1...CDNNT*
A209563 u x....1....0....x....x.....1...CCFT^
A209564 v x....1....0....x....x.....1...CFN^
A209565 u x....1....0....x....2x....1...CC^
A209566 v x....1....0....x....2x....1...BC^
A209567 u x....1....0....x....x+1...1...CNT*
A209568 v x....1....0....x....x+1...1...NNS*
A209569 u x....1....0....2x...1.....1...CNO*
A209570 v x....1....0....2x...1.....1...DNN*
A209571 u x....1....0....2x...x.....1...CCS^
A209572 v x....1....0....2x...x.....1...CN^
A209573 u x....1....0....2x...x+1...1...CNS
A209574 v x....1....0....2x...x+1...1...NO
A209575 u x....1....0....2x...2x....1...CC
A209576 v x....1....0....2x...2x....1...C
A209577 u x....1....0....x+1..1.....1...CNNT
A209578 v x....1....0....x+1..1.....1...CNN
A209579 u x....1....0....x+1..x.....1...CNNT
A209580 v x....1....0....x+1..x.....1...NN*
A209581 u x....1....0....x+1..2x....1...CN
A209582 v x....1....0....x+1..2x....1...BN
A209583 u x....1....0....x+1..x+1...1...CT*
A209584 v x....1....0....x+1..x+1...1...CN*
A121462 u x....x....0....x....x+1...0...BCFFNZ
A208341 v x....x....0....x....x+1...0...BCFFN
A209687 u x....x....0....2x...x+1...0...BCNZ
A208339 v x....x....0....2x...x+1...0...BCN
A115241 u x....x....0....1....1.....1...CDNZ*
A209688 v x....x....0....1....1.....1...DDN*
A209689 u x....x....0....1....x.....1...FNZ^
A209690 v x....x....0....1....x.....1...FN^
A209691 u x....x....0....1....2x....1...BCZ^
A209692 v x....x....0....1....2x....1...BCC^
A209693 u x....x....0....1....x+1...1...NNZ*
A209694 v x....x....0....1....x+1...1...CN*
A209697 u x....x....0....x....x+1...1...BNZ
A209698 v x....x....0....x....x+1...1...BNT
A209699 u x....x....0....2x...1.....1...BNNZ
A209700 v x....x....0....2x...1.....1...BDN
A209701 u x....x....0....2x...x+1...1...NZ
A209702 v x....x....0....2x...x+1...1...N
A209703 u x....x....0....x+1..1.....1...FNTZ
A209704 v x....x....0....x+1..1.....1...FNNT
A209705 u x....x....0....x+1..x+1...1...BNZ*
A209706 v x....x....0....x+1..x+1...1...BCN*
A209695 u x....x+1..0....2x...x+1...0...ACN*
A209696 v x....x+1..0....2x...x+1...0...CDN*
A209830 u x....x+1..0....x+1..2x....0...ACF
A209831 v x....x+1..0....x+1..2x....0...BCF*
A209745 u x....x+1..0....x+1..x+1...0...ABF*
A209746 v x....x+1..0....x+1..x+1...0...BFZ*
A209747 u x....x+1..0....1....1.....1...ADE*
A209748 v x....x+1..0....1....1.....1...DEO
A209749 u x....x+1..0....1....x.....1...ANN*
A209750 v x....x+1..0....1....x.....1...CNO
A209751 u x....x+1..0....1....2x....1...ABN*
A209752 v x....x+1..0....1....2x....1...BN
A209753 u x....x+1..0....1....x+1...1...AN*
A209754 v x....x+1..0....1....x+1...1...NT*
A209755 u x....x+1..0....x....1.....1...AFN
A209756 v x....x+1..0....x....1.....1...FNO*
A209759 u x....x+1..0....x....2x....1...ACF^
A209760 v x....x+1..0....x....2x....1...CF^*
A209761 u x....x+1..0....x.....x+1..1...ABNS*
A209762 v x....x+1..0....x.....x+1..1...BNS*
A209763 u x....x+1..0....2x....1....1...ABN*
A209764 v x....x+1..0....2x....1....1...BNN
A209765 u x....x+1..0....2x....x....1...ACF^*
A209766 v x....x+1..0....2x....x....1...CF^
A209767 u x....x+1..0....2x....x+1..1...AN*
A209768 v x....x+1..0....2x....x+1..1...N*
A209769 u x....x+1..0....x+1...1....1...AF*
A209770 v x....x+1..0....x+1...1....1...FN
A209771 u x....x+1..0....x+1...x....1...ABN*
A209772 v x....x+1..0....x+1...x....1...BN*
A209773 u x....x+1..0....x+1...2x...1...AF
A209774 v x....x+1..0....x+1...2x...1...FN*
A209775 u x....x+1..0....x+1...x+1..1...AB*
A209776 v x....x+1..0....x+1...x+1..1...BC*
A210033 u 1....1....1....1.....x....1...BCN
A210034 v 1....1....1....1.....x....1...BCDFN
A210035 u 1....1....1....1.....2x...1...BBF
A210036 v 1....1....1....1.....2x...1...BBFF
A210037 u 1....1....1....1.....x+1..1...BCFFN
A210038 v 1....1....1....1.....x+1..1...BCFFN
A210039 u 1....1....1....x.....1....1...BCOT
A210040 v 1....1....1....x.....1....1...BCEN
A210042 u 1....1....1....x.....x....1...BCDEOT*
A124927 v 1....1....1....x.....x....1...BCDET*
A210041 u 1....1....1....x.....2x...1...BFO
A209758 v 1....1....1....x.....2x...1...BCFO
A210187 u 1....1....1....x.....x+1..1...DTF*
A210188 v 1....1....1....x.....x+1..1...DNF*
A210189 u 1....1....1....2x....1....1...BT
A210190 v 1....1....1....2x....1....1...BN
A210191 u 1....1....1....2x....x....1...CO*
A210192 v 1....1....1....2x....x....1...CCO*
A210193 u 1....1....1....2x....x+1..1...CPT
A210194 v 1....1....1....2x....x+1..1...CN
A210195 u 1....1....1....2x....2x...1...BOPT*
A210196 v 1....1....1....2x....2x...1...BCC*
A210197 u 1....1....1....x+1...1....1...BCOT
A210198 v 1....1....1....x+1...1....1...BCEN
A210199 u 1....1....1....x+1...x....1...DFT
A210200 v 1....1....1....x+1...x....1...DFO*
A210201 u 1....1....1....x+1...2x...1...BFP
A210202 v 1....1....1....x+1...2x...1...BF
A210203 u 1....1....1....x+1...x+1..1...BDOP
A210204 v 1....1....1....x+1...x+1..1...BCDN*
A210211 u x....1....1....1.....2x...1...BCFN
A210212 v x....1....1....1.....2x...1...BFN
A210213 u x....1....1....1.....x+1..1...CFFN
A210214 v x....1....1....1.....x+1..1...CFFO
A210215 u x....1....1....x.....x....1...BCDFT^
A210216 v x....1....1....x.....x....1...BCFO^
A210217 u x....1....1....x.....2x...1...CDF^
A210218 v x....1....1....x.....2x...1...BCF^
A210219 u x....1....1....x.....x+1..1...CNSTF*
A210220 v x....1....1....x.....x+1..1...FNNT*
A104698 u x....1....1....2x......1..1...CENS*
A210220 v x....1....1....2x....x+1..1...DNNT*
A210223 u x....1....1....2x....x....1...CD^
A210224 v x....1....1....2x....x....1...CO^
A210225 u x....1....1....2x....x+1..1...CNP
A210226 v x....1....1....2x....x+1..1...NOT
A210227 u x....1....1....2x....2x...1...CDP^
A210228 v x....1....1....2x....2x...1...C^
A210229 u x....1....1....x+1...1....1...CFNN
A210230 v x....1....1....x+1...1....1...CCN
A210231 u x....1....1....x+1...x....1...CNT
A210232 v x....1....1....x+1...x....1...NN*
A210233 u x....1....1....x+1...2x...1...CNP
A210234 v x....1....1....x+1...2x...1...BN
A210235 u x....1....1....x+1...x+1..1...CCFPT*
A210236 v x....1....1....x+1...x+1..1...CFN*
A124927 u x....x....1....1.....1....1...BCDEET*
A210042 v x....1....1....x+1...x+1..1...BDEOT*
A210216 u x....x....1....1.....x....1...BCFO^
A210215 v x....x....1....1.....x....1...BCDFT^
A210549 u x....x....1....1.....2x...1...BCF^
A210550 v x....x....1....1.....2x...1...BDF^
A172431 u x....x....1....1.....x+1..1...CEFN*
A210551 v x....x....1....1.....x+1..1...CFOT*
A210552 u x....x....1....x.....1....1...BBCFNO
A210553 v x....x....1....x.....1....1...BNNFB
A208341 u x....x....1....x.....x+1..1...BCFFN
A210554 v x....x....1....x.....x+1..1...BNFFT
A210555 u x....x....1....2x....1....1...BCNN
A210556 v x....x....1....2x....1....1...BENP
A210557 u x....x....1....2x....x+1..1...CNP
A210558 v x....x....1....2x....x+1..1...N
A210559 u x....x....1....x+1...1....1...CEF
A210560 v x....x....1....x+1...1....1...OFNS
A210561 u x....x....1....x+1...x....1...BCNP^
A210562 v x....x....1....x+1...x....1...BDP*^
A210563 u x....x....1....x+1...2x...1...CFP^
A210564 v x....x....1....x+1...2x...1...DF^
A013609 u x....x....1....x+1...x+1..1...BCEPT*
A209757 v x....x....1....x+1...x+1..1...BCOS*
A209819 u x....2x...1....x+1...x....1...CFN^
A209820 v x....2x...1....x+1...x....1...DF^
A209996 u x....2x...1....x+1...2x...1...CP^
A209998 v x....2x...1....x+1...2x...1...DP^
A209999 u x....x+1..1....1.....x+1..1...FN*
A210287 v x....x+1..1....1.....x+1..1...CFT*
A210565 u x....x+1..1....x.....1....1...FNT*
A210595 v x....x+1..1....x.....1....1...FNNT
A210598 u x....x+1..1....x+1...2x...1...FN*
A210599 v x....x+1..1....x+1...2x...1...FN
A210600 u x....x+1..1....x+1...x+1..1...BF*
A210601 v x....x+1..1....x+1...x+1..1...BF*
A210597 u 2x...1....1....x+1...1....1...BF
A210601 v 2x...1....1....x+1...1....1...BFN*
A210603 u 2x...1....1....x+1...x+1..1...BF
A210738 v 2x...1....1....x+1...x+1..1...CBF*
A210739 u 2x...x....1....x+1...x....1...CF^
A210740 v 2x...x....1....x+1...x....1...DF*^
A210741 u 2x...x....1....x+1...x+1..1...BCFO
A210742 v 2x...x....1....x+1...x+1..1...CFO*
A210743 u 2x...x+1..1....x+1...1....1...F
A210744 v 2x...x+1..1....x+1...1....1...FN
A210747 u 2x...x+1..1....x+1...x+1..1...FF
A210748 v 2x...x+1..1....x+1...x+1..1...CFF*
A210749 u x+1..1....1....x+1...2x...1...BCF
A210750 v x+1..1....1....x+1...2x...1...BF
A210751 u x+1..x....1....x+1...2x...1...FNT
A210752 v x+1..x....1....x+1...2x...1...FN
A210753 u x+1..x....1....x+1...x+1..1...BNZ*
A210754 v x+1..x....1....x+1...x+1..1...BCT*
A210755 u x+1..2x...1....x+1...x+1..1...N*
A210756 v x+1..2x...1....x+1...x+1..1...CT*
A210789 u 1....x....0....x+2...x-1..0...CFFN
A210790 v 1....x....0....x+2...x-1..0...CEFF
A210791 u 1....x....0....x-1...x+2..0...CFNP
A210792 v 1....x....0....x-1...x+2..0...CF
A210793 u 1....x+1..0....x+2...x-1..0...CFNP
A210794 v 1....x+1..0....x+2...x-1..0...FPP
A210795 u 1....x....1....x+2...x-1..0...FN
A210796 v 1....x....1....x+2...x-1..0...FO
A210797 u 1....x....0....x+2...x-1..1...CF
A210798 v 1....x....0....x+2...x-1..1...F
A210799 u 1....x+1..1....x+2...x-1..0...FN
A210800 v 1....x+1..1....x+2...x-1..0...F
A210801 u 1....x+1..1....x+2...x-1..1...FN
A210802 v 1....x+1..1....x+2...x-1..1...F
A210803 u 1....x....0....x-1...x+3..0...F*
A210804 v 1....x....0....x-1...x+3..0...F*
A210805 u 1....x....0....x+2...x-1.-1...CFFN
A210806 v 1....x....0....x+2...x-1.-1...FF
A210858 u 1....x....0....x+n...x....0...CFT*
A210859 v 1....x....0....x+n...x....0...FN*
A210860 u 1....x+1..0....x+n...x....0...F
A210861 v 1....x+1..0....x+n...x....0...F*
A210862 u 1....x....1....x+n-1.x....0...FN
A210863 v 1....x....1....x+n-1.x....0...FS
A210864 u 1....x....1....x+n...x....0...FN
A210865 v 1....x....1....x+n...x....0...FT
A210866 u 1....x....0....x+n...x...-x...CFT
A210867 v 1....x....0....x+n...x...-x...FN
A210868 u 1....x....0....x+1...x-1..0...BCFN
A210869 v 1....x....0....x+1...x-1..0...BBCFNZ
A210870 u 1....x....0....x+1...x-1..1...CFFN
A210871 v 1....x....0....x+1...x-1..1...CFF
A210872 u x....1...-1....x.....x....1...BDFZ^
A210873 v x....1...-1....x.....x....1...BCFN^
A210876 u x....1....1....x.....x....x...BCCF^
A210877 v x....1....1....x.....x....x...BDFNZ^
A210878 u x....2x...0....x+1...x....1...DFZ^
A210879 v x....2x...0....x+1...x....1...FC*^
Some of these triangles have irregular row lengths, making it difficult to retrieve individual rows/columns/diagonals without actually computing the recurrence. - Georg Fischer, Sep 04 2021

Examples

			First five rows:
1
1...1
1...3...1
1...5...4...1
1...7...9...5...1
First five polynomials u(n,x):
1
1 + x
1 + 3x + x^2
1 + 5x + 4x^2 + x^3
1 + 7x + 9x^2 + 5x^3 + x^4

Crossrefs

Cf. A029653, A208508, A208509.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A208510 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A029653 *)
  • Python
    from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1
def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 27 2017

Formula

u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Also, u(n,x)=(x+1)*u(n-1,x)+x for n>2, with u(n,2)=x+1.

Extensions

Corrected by Philippe Deléham, Apr 10 2012
Corrections and additions by Clark Kimberling, May 09 2012
Corrections in the overview by Georg Fischer, Sep 04 2021

A028326 Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).

Original entry on oeis.org

2, 2, 2, 2, 4, 2, 2, 6, 6, 2, 2, 8, 12, 8, 2, 2, 10, 20, 20, 10, 2, 2, 12, 30, 40, 30, 12, 2, 2, 14, 42, 70, 70, 42, 14, 2, 2, 16, 56, 112, 140, 112, 56, 16, 2, 2, 18, 72, 168, 252, 252, 168, 72, 18, 2, 2, 20, 90, 240, 420, 504, 420, 240, 90, 20, 2, 2, 22, 110, 330, 660, 924, 924, 660, 330, 110, 22, 2
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Comments

Also number of binary vectors of length n+1 with k+1 runs (1 <= k <= n).
If the last two entries in each row are removed and 0 replaces the entries in a checkerboard pattern, we obtain
2;
0, 6;
2, 0, 12;
0, 10, 0, 20;
2, 0, 30, 0, 30;
0, 14, 0, 70, 0, 42;
2, 0, 56, 0, 140, 0, 56;
0, 18, 0, 168, 0, 252, 0, 72;
...
This plays the same role of recurrence coefficients for second differences of polynomials as triangle A074909 plays for the first differences. - R. J. Mathar, Jul 03 2013
From Roger Ford, Jul 06 2023: (Start)
T(n,k) = the number of closed meanders with n top arches, n+1 exterior arches and with k = the number of arches of length 1 - (n+1).
Example of closed meanders with 4 top arches and 5 exterior arches:
exterior arches are top arches or bottom arches without a covering arch
/\ = top arch length 1, \/ = bottom arch length 1
/ \ Top: /\=3 / \ / \ Top: /\=2
/\ / /\ \ /\ / /\ \ / /\ \
\ \/ / \ \/ / Bottom: \/=2 \/ \ \/ / \/ Bottom: /\=3
\/ \/ k=5-5=0 \/ k=5-5=0 T(4,0) = 2
____
/ \ Top: /\=3 / \ Top: /\=3
/\ / /\ /\ \ / /\ \ /\ /\
\ \/ / \/ \/ Bottom: \/=3 \/ \ \/ \/ / Bottom: \/=3
\/ k=6-5=1 \____/ k=6-5=1
____
/ \ Top: /\=3 / \ Top: /\=3
/ /\ /\ \ /\ /\ /\ / /\ \
\/ \/ \ \/ / Bottom: \/=3 \ \/ \/ / \/ Bottom: \/=3
\/ k=6-5=1 \____/ k=6-5=1 T(4,1) = 4
________
/ \ Top: /\=3
/ /\ /\ /\ \ /\ /\ /\ /\ Top: /\=4
\/ \/ \/ \/ Bottom: \/=4 \ \/ \/ \/ / Bottom: ||=3
k=7-5=2 \________/ k=7-5=2 T(4,2) = 2.
(End)

Examples

			Triangle begins:
  2;
  2,  2;
  2,  4,   2;
  2,  6,   6,   2;
  2,  8,  12,   8,   2;
  2, 10,  20,  20,  10,    2;
  2, 12,  30,  40,  30,   12,    2;
  2, 14,  42,  70,  70,   42,   14,    2;
  2, 16,  56, 112, 140,  112,   56,   16,   2;
  2, 18,  72, 168, 252,  252,  168,   72,  18,   2;
  2, 20,  90, 240, 420,  504,  420,  240,  90,  20,   2;
  2, 22, 110, 330, 660,  924,  924,  660, 330, 110,  22,  2;
  2, 24, 132, 440, 990, 1584, 1848, 1584, 990, 440, 132, 24, 2;

References

  • I. Goulden and D. Jackson, Combinatorial Enumeration, John Wiley and Sons, 1983, page 76.

Links

Crossrefs

Cf. A007318, A028327, A028328, A028329, A028330, A028331, A028332, A124927, A134058.

Programs

  • Haskell
    a028326 n k = a028326_tabl !! n !! k
a028326_row n = a028326_tabl !! n
a028326_tabl = iterate
   (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [2]
-- Reinhard Zumkeller, Mar 12 2012
  • Magma
    [2*Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 27 2021
  • Maple
    T := proc(n, k) if k=0 then 2 elif k>n then 0 else T(n-1, k)+T(n-1, k-1) fi end:
for n from 0 to 13 do seq(T(n, k), k=0..n) od; # Zerinvary Lajos, Dec 16 2006
  • Mathematica
    Table[2*Binomial[n, k], {n, 0, 11}, {k, 0, n}]//Flatten (* Robert G. Wilson v, Mar 05 2012 *)
  • PARI
    T(n,k) = 2*binomial(n,k) \\ Charles R Greathouse IV, Feb 07 2017
  • Python
    from sympy import binomial
def T(n, k):
    return 2*binomial(n, k)
for n in range(21): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Apr 29 2017
  • Sage
    flatten([[2*binomial(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 27 2021

Formula

G.f. for the number of length n binary words with k runs: (1-x+x*y)/(1-x-x*y) [Goulden and Jackson]. - Geoffrey Critzer, Mar 04 2012

Extensions

More terms from Donald Manchester, Jr. (s1199170(AT)cedarnet.cedarville.edu)

A100320 A Catalan transform of (1 + 2*x)/(1 - 2*x).

Original entry on oeis.org

1, 4, 12, 40, 140, 504, 1848, 6864, 25740, 97240, 369512, 1410864, 5408312, 20801200, 80233200, 310235040, 1202160780, 4667212440, 18150270600, 70690527600, 275693057640, 1076515748880, 4208197927440, 16466861455200, 64495207366200, 252821212875504, 991837065896208
Offset: 0

Views

Author

Paul Barry, Nov 14 2004

Keywords

Comments

A Catalan transform of (1 + 2*x)/(1 - 2*x) under the mapping g(x) -> g(x*c(x)). (Here c(x) is the g.f. of A000108.) The original sequence can be retrieved by g(x) -> g(x*(1-x)).
Hankel transform is A144704. - Paul Barry, Sep 19 2008
Central terms of the triangle in A124927. - Reinhard Zumkeller, Mar 04 2012

Links

Crossrefs

Cf. A000108, A010684, A028329, A039599, A095660, A124927, A132046, A144704.

Programs

Formula

G.f.: (1 + 2*x*c(x))/(1 - 2*x*c(x)), where c(x) is the g.f. of A000108.
a(n) = 4*binomial(2*n-1, n) - 3*0^n.
a(n) = binomial(2*n, n)*(4*2^(n-1) - 0^n)/2^n.
a(n) = Sum_{j=0..n} Sum_{k=0..n} C(2*n, n-k)*((2*k + 1)/(n + k + 1))*C(k, j)*(-1)^(j-k)*(4*2^(j-1) - 0^j).
a(n) = A028329(n), n > 0. - R. J. Mathar, Sep 02 2008
a(n) = T(2*n,n), where T(n,k) = A132046(n,k). - Paul Barry, Sep 19 2008
a(n) = Sum_{k=0..n} A039599(n,k)*A010684(k). - Philippe Deléham, Oct 29 2008
a(n) = A095660(2*n,n) for n > 0. - Reinhard Zumkeller, Apr 08 2012
G.f.: G(0) - 1, where G(k) = 1 + 1/(1 - 2*x*(2*k + 1)/(2*x*(2*k + 1) + (k + 1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
a(n) = [x^n] (1 + 2*x)/(1 - x)^(n+1). - Ilya Gutkovskiy, Oct 12 2017
a(n) = 2*(2*n-1)*a(n-1)/n. - G. C. Greubel, Feb 01 2023
E.g.f.: 2*exp(2*x)*BesselI(0, 2*x) - 1. - Stefano Spezia, May 11 2024

Extensions

Incorrect connection with A046055 deleted by N. J. A. Sloane, Jul 08 2009

A124928 Triangle read by rows: T(n,0) = 1, T(n,k) = 3*binomial(n,k) if k>=0 (0<=k<=n).

Original entry on oeis.org

1, 1, 3, 1, 6, 3, 1, 9, 9, 3, 1, 12, 18, 12, 3, 1, 15, 30, 30, 15, 3, 1, 18, 45, 60, 45, 18, 3, 1, 21, 63, 105, 105, 63, 21, 3, 1, 24, 84, 168, 210, 168, 84, 24, 3, 1, 27, 108, 252, 378, 378, 252, 108, 27, 3, 1, 30, 135, 360, 630, 756, 630, 360, 135, 30, 3
Offset: 0

Views

Author

Gary W. Adamson, Nov 12 2006

Keywords

Comments

Row sums = A033484: (1, 4, 10, 22, 46, 94...); 3*2^n - 2.
Analogous triangle using (1,2,2,2...) as the main diagonal of M = A124927.
Except for the first column, entries in the Pascal triangle are tripled.

Examples

			First few rows of the triangle are:
  1;
  1,  3;
  1,  6,  3;
  1,  9,  9,  3;
  1, 12, 18, 12,  3;
  1, 15, 30, 30, 15,  3;
  1, 18, 45, 60, 45, 18, 3;
...

Links

Crossrefs

Cf. A033484, A124927.

Programs

  • GAP
    T:= function(n,k)
    if k=0 then return 1;
    else return 3*Binomial(n,k);
    fi;  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 19 2019
  • Magma
    [k eq 0 select 1 else 3*Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 19 2019
  • Maple
    T:=proc(n,k) if k=0 then 1 else 3*binomial(n,k) fi end: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
  • Mathematica
    Flatten[Table[If[k==0,1,3*Binomial[n,k]],{n,0,20},{k,0,n}]] (* Harvey P. Dale, Oct 19 2013 *)
  • PARI
    T(n,k) = if(k==0, 1, 3*binomial(n,k)); \\ G. C. Greubel, Nov 19 2019
  • Sage
    def T(n, k):
    if (k==0): return 1
    else: return 3*binomial(n,k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 19 2019

Formula

G.f.: G(t,z) = 3/(1-(1+t)*z) - 2/(1-z).

Extensions

Edited by N. J. A. Sloane, Nov 29 2006

A136521 Triangle read by rows: (1, 2, 2, 2, ...) on the main diagonal and the rest zeros.

Original entry on oeis.org

1, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2
Offset: 0

Views

Author

Gary W. Adamson, Jan 02 2008

Keywords

Examples

			First few rows of the triangle are:
  1;
  0, 2;
  0, 0, 2;
  0, 0, 0, 2;
  0, 0, 0, 0, 2;
  ...

Links

Crossrefs

Cf. A001263, A007318, A040000, A124927, A134058, A136522.

Programs

  • Mathematica
    Table[2*Boole[k==n] -Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 03 2021 *)
  • Sage
    flatten([[2*bool(k==n) -bool(n==0) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 03 2021

Formula

By columns, (1, 0, 0, 0, ...) in leftmost column; all others are (2, 0, 0, 0, ...).
By rows, row 1 = 1, others = (n-1) zeros followed by "2".
A007318(n,k) * T(n,k) = A124927(n,k).
T(n,k) * A007318(n,k) = A134058(n,k).
A001263(n,k) * T(n,k) = A136522(n,k).
From G. C. Greubel, May 03 2021: (Start)
T(n, k) = 2*[k=n] - [n=0].
Sum_{k=0..n} T(n, k) = A040000(n). (End)

Extensions

More terms added by G. C. Greubel, May 03 2021
Showing 1-8 of 8 results.

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