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 1-6 of 6 results.

A106516 A Pascal-like triangle based on 3^n.

Original entry on oeis.org

1, 3, 1, 9, 4, 1, 27, 13, 5, 1, 81, 40, 18, 6, 1, 243, 121, 58, 24, 7, 1, 729, 364, 179, 82, 31, 8, 1, 2187, 1093, 543, 261, 113, 39, 9, 1, 6561, 3280, 1636, 804, 374, 152, 48, 10, 1, 19683, 9841, 4916, 2440, 1178, 526, 200, 58, 11, 1, 59049, 29524, 14757, 7356, 3618, 1704, 726, 258, 69, 12, 1
Offset: 0

Views

Author

Paul Barry, May 05 2005

Keywords

Comments

Row sums are A027649. Antidiagonal sums are A106517.
From Wolfdieter Lang, Jan 09 2015: (Start)
Alternating row sums give A025192. The A-sequence of this Riordan lower triangular matrix is [1, 1, repeat(0, )] (leading to the Pascal recurrence for T(n,k) for n >= k >= 1. The Z-sequence is [3, repeat(0, )] (leading to the recurrence T(n,0) = 3*T(n-1,0), n >= 1. For A- and Z-sequences see the W. Lang link under A006232.
The inverse of this Riordan matrix is Tinv = ((1 - 2*x)/(1 + x), x/(1 + x)) given as a signed version of A093560: Tinv(n,m) = (-1)^(n-m)*A093560(n,m). (End)

Examples

			The triangle T(n,k) begins:
n\k     0     1     2    3    4    5   6   7  8  9 10 ...
0:      1
1:      3     1
2:      9     4     1
3:     27    13     5    1
4:     81    40    18    6    1
5:    243   121    58   24    7    1
6:    729   364   179   82   31    8   1
7:   2187  1093   543  261  113   39   9   1
8:   6561  3280  1636  804  374  152  48  10  1
9:  19683  9841  4916 2440 1178  526 200  58 11  1
10: 59049 29524 14757 7356 3618 1704 726 258 69 12  1
... reformatted and extended. - _Wolfdieter Lang_, Jan 06 2015
----------------------------------------------------------
With the array M(k) as defined in the Formula section, the infinite product M(0)*M(1)*M(2)*... begins
/ 1        \/1           \/1        \       /1         \
| 3  1     ||0  1        ||0 1      |      | 3  1      |
| 9  4 1   ||0  3  1     ||0 0 1    |... = | 9  7  1   |
|27 13 5 1 ||0  9  4 1   ||0 0 3 1  |      |27 37 12 1 |
|...       ||0 27 13 5 1 ||0 0 9 4 1|      |...        |
|...       ||...         ||...      |      |...        |
= A143495. - _Peter Bala_, Dec 23 2014

Crossrefs

Columns 1, 2, 3, 4, 5: A003462, A000340, A052150, A097786, A097787.
Cf. A055248, A143495.

Programs

  • Mathematica
    a106516[n_] := Block[{a, k},
a[x_] := Flatten@ Last@ Reap[For[k = -1, k < x, Sow[Binomial[x, k] +
2 Sum[3^(i - 1)*Binomial[x - i, k], {i, 1, x}]], k++]]; Flatten@Array[a, n, 0]]; a106516[11] (* Michael De Vlieger, Dec 23 2014 *)

Formula

Riordan array (1/(1-3x), x/(1-x)); Number triangle T(n, 0)=A000244(n), T(n, k)=T(n-1, k-1)+T(n-1, k); T(n, k)=sum{j=0..n, binomial(n, k+j)2^j}.
From Peter Bala, Jul 16 2013: (Start)
T(n,k) = binomial(n,k) + 2*sum {i = 1..n} 3^(i-1)*binomial(n-i,k).
O.g.f.: (1 - t)/( (1 - 3*t)*(1 - (1 + x)*t) ) = 1 + (3 + x)*t + (9 + 4*x + x^2)*t^2 + ....
The n-th row polynomial R(n,x) = 1/(x - 2)*( x*(x + 1)^n - 2*3^n ). (End)
Closed-form formula for arbitrary left and right borders of Pascal-like triangle see A228196. - Boris Putievskiy, Aug 19 2013
T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 3*T(n-2,k-1), T(0,0)=1, T(1,0)=3, T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Dec 26 2013
From Peter Bala, Dec 23 2014: (Start)
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(27 + 13*x + 5*x^2/2! + x^3/3!) = 27 + 40*x + 58*x^2/2! + 82*x^3/3! + 113*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ).
Let M denote the present triangle. For k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0 M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to A143495 (but with a different offset). See the Example section. Cf. A055248. (End)
n-th row polynomial R(n, x) = (2*3^n - x*(1 + x)^n)/(2 - x). - Peter Bala, Mar 05 2025

A097786 a(n) = 3*a(n-1) + C(n+3,3) for n > 0; a(0)=1.

Original entry on oeis.org

1, 7, 31, 113, 374, 1178, 3618, 10974, 33087, 99481, 298729, 896551, 2690108, 8070884, 24213332, 72640812, 217923405, 653771355, 1961315395, 5883947725, 17651844946, 52955536862, 158866612886, 476599841258, 1429799526699
Offset: 0

Views

Author

Paul Barry, Aug 24 2004

Keywords

Comments

Partial sums of A052150.

Links

Programs

  • Mathematica
    RecurrenceTable[{a[0]==1,a[n]==3a[n-1]+Binomial[n+3,3]},a,{n,30}] (* or *) LinearRecurrence[{7,-18,22,-13,3},{1,7,31,113,374},31] (* Harvey P. Dale, Nov 26 2011 *)

Formula

G.f.: 1/((1-3*x)*(1-x)^4);
a(n) = 3^(n+4)/16 - (4*n^3 + 42*n^2 + 152*n + 195)/48;
a(n) = Sum_{k=0..n} binomial(n+4, k+4)*2^k.
a(n) = 7*a(n-1) - 18*a(n-2) + 22*a(n-3) - 13*a(n-4) + 3*a(n-5); a(0) = 1, a(1)=7, a(2)=31, a(3)=113, a(4)=374. - Harvey P. Dale, Nov 26 2011

A368534 a(n) = Sum_{k=1..n} binomial(k+1,2) * n^(n-k).

Original entry on oeis.org

0, 1, 5, 24, 146, 1215, 13431, 186816, 3130436, 61291125, 1371742105, 34522712136, 964626945558, 29621465864627, 991330604373851, 35906022352657920, 1399219698628043016, 58367293868445147657, 2594796705962971336125, 122463905297217627859000
Offset: 0

Views

Author

Seiichi Manyama, Dec 29 2023

Keywords

Crossrefs

Cf. A062805, A368535.
Cf. A002662, A052150, A052161.
Cf. A023037, A062806, A068475.

Programs

  • Mathematica
    Table[Sum[Binomial[k+1,2]n^(n-k),{k,n}],{n,0,20}] (* Harvey P. Dale, May 14 2025 *)
  • PARI
    a(n) = sum(k=1, n, binomial(k+1, 2)*n^(n-k));

Formula

a(n) = [x^n] x/((1-n*x) * (1-x)^3).
a(n) = n * (2*n^(n+1) - n^3 - n^2 + n - 1)/(2 * (n-1)^3) for n > 1.

A229611 Expansion of 1/((1-x)^3*(1-11x)).

Original entry on oeis.org

1, 14, 160, 1770, 19485, 214356, 2357944, 25937420, 285311665, 3138428370, 34522712136, 379749833574, 4177248169405, 45949729863560, 505447028499280, 5559917313492216, 61159090448414529, 672749994932559990, 7400249944258160080, 81402749386839761090
Offset: 0

Views

Author

Yahia Kahloune, Sep 26 2013

Keywords

Comments

This sequence was chosen to illustrate a method of matching generating functions and closed-form solutions: The general term associated with the generating function 1/((1-s*x)^3*(1-r*x)) with r>s>=1 is a(n) = [r^(n+3) - s^(n+1)*(s^2 + (r-s)*s*binomial(n+3,1) +(r-s)^2*binomial(n+3,2))] / (r-s)^3 .

Examples

			a(3) = (11^6 - (50*3^2+260*3 + 331))/1000 = 1770 .

Links

Crossrefs

Cf. A002662, A052150, A052161, A052244, A052262.

Programs

  • Magma
    [(11^(n+3) - (50*n^2 + 260*n + 331))/1000: n in [0..25]]; // Vincenzo Librandi, Sep 27 2013
  • Mathematica
    CoefficientList[Series[1/((1 - x)^3 (1 - 11 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Sep 27 2013 *)
LinearRecurrence[{14,-36,34,-11},{1,14,160,1770},30] (* Harvey P. Dale, Apr 09 2016 *)

Formula

a(n) = (11^(n+3) - (1 + 10*C(n+3,1) + 100*C(n+3,2)))/1000 = (11^(n+3) - (50*n^2 + 260*n + 331))/1000.
a(n) = 14*a(n-1) -36*a(n-2) +34*a(n-3) -11*a(n-4). - Vincenzo Librandi, Sep 27 2013

A368528 a(n) = Sum_{k=1..n} k^2 * 3^(n-k).

Original entry on oeis.org

0, 1, 7, 30, 106, 343, 1065, 3244, 9796, 29469, 88507, 265642, 797070, 2391379, 7174333, 21523224, 64569928, 193710073, 581130543, 1743391990, 5230176370, 15690529551, 47071589137, 141214767940, 423644304396, 1270932913813, 3812798742115
Offset: 0

Views

Author

Seiichi Manyama, Dec 28 2023

Keywords

Links

Crossrefs

Cf. A000290, A000330, A047520, A368529.
Cf. A052150, A368524.

Programs

  • PARI
    a(n) = sum(k=1, n, k^2*3^(n-k));

Formula

G.f.: x * (1+x)/((1-3*x) * (1-x)^3).
a(n) = 6*a(n-1) - 12*a(n-2) + 10*a(n-3) - 3*a(n-4).
a(n) = A052150(n-1) + A052150(n-2) for n > 1.
a(n) = (3^(n+1) - (n^2 + 3*n + 3))/2.
a(0) = 0; a(n) = 3*a(n-1) + n^2.

A368506 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * binomial(j+k-1,j).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 6, 11, 4, 0, 1, 8, 24, 26, 5, 0, 1, 10, 42, 82, 57, 6, 0, 1, 12, 65, 188, 261, 120, 7, 0, 1, 14, 93, 360, 787, 804, 247, 8, 0, 1, 16, 126, 614, 1870, 3204, 2440, 502, 9, 0, 1, 18, 164, 966, 3810, 9476, 12900, 7356, 1013, 10, 0
Offset: 0

Views

Author

Seiichi Manyama, Dec 27 2023

Keywords

Examples

			Square array begins:
  1, 1,   1,    1,     1,     1,      1, ...
  0, 2,   4,    6,     8,    10,     12, ...
  0, 3,  11,   24,    42,    65,     93, ...
  0, 4,  26,   82,   188,   360,    614, ...
  0, 5,  57,  261,   787,  1870,   3810, ...
  0, 6, 120,  804,  3204,  9476,  23112, ...
  0, 7, 247, 2440, 12900, 47590, 139134, ...

Crossrefs

Columns k=0..3 give A000007, A000027(n+1), A125128(n+1), A052150.
Main diagonal gives A293574.
Cf. A008949, A368487.

Programs

  • PARI
    T(n, k) = sum(j=0, n, k^(n-j)*binomial(j+k-1, j));

Formula

G.f. of column k: 1/((1-k*x) * (1-x)^k).
Showing 1-6 of 6 results.

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