A108284 -id:A108284 - cp's OEIS frontend

A108283 Triangle read by rows, generated from (..., 3, 2, 1).

1, 1, 3, 1, 5, 6, 1, 7, 17, 10, 1, 9, 34, 49, 15, 1, 11, 57, 142, 129, 21, 1, 13, 86, 313, 547, 321, 28, 1, 15, 121, 586, 1593, 2005, 769, 36, 1, 17, 162, 985, 3711, 7737, 7108, 1793, 45, 1, 19, 209, 1534, 7465, 22461, 36409, 24604, 4097, 55, 1, 21, 262, 2257, 13539, 54121, 131836, 167481, 83653, 9217, 66

Offset: 1

Gary W. Adamson, May 30 2005

Inverse binomial transforms of each column form the rows of A108284. Rightmost diagonal = triangular numbers, (A000217); while diagonals going to the left from (1, 3, 6, ...) are A000337 starting with 1: (1, 5, 17, 49, ...); A014915: (1, 7, 34, 142, ...); A014916: (1, 9, 57, ...); A014917: (1, 11, 86, ...).

			4th column = 10, 49, 142, 313, ... = f(x), x = 1, 2, 3; 4x^3 + 3x^2 + 2x + 1. f(3) = 142.
First few rows of the triangle:
  1;
  1,  3;
  1,  5,  6;
  1,  7, 17,  10;
  1,  9, 34,  49,  15;
  1, 11, 57, 142, 129, 21;
  ...

Cf. A059045, A108284, A000217, A000337, A014915, A014916, A014917.

A108283 := proc(n,k)
    local x ;
    x := n-k+1 ;
    add( i*x^(i-1),i=1..k) ;
end proc:
seq(seq( A108283(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Sep 14 2016

T[, 1] := 1; T[n, n_] := n (n + 1)/2; T[n_, k_] := (1 - (n - k + 1)^k*(k^2 - k*n + 1))/(n - k)^2; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 13 2016 *)

n-th column = f(x), x = 1, 2, 3; n*x^(n-1) + (n-1)*x^(n-2) + (n-3)*x^(n-3) + ... + 1.

T(n,k) = (1+ (n-k+1)^k*(n*k-k^2-1))/ (n-k)^2, n>k. - Jean-François Alcover, Sep 13 2016

More terms from Jean-François Alcover, Sep 13 2016

A108285 Triangle read by rows, generated from (1, 2, 3, ...).

Original entry on oeis.org

1, 1, 3, 1, 4, 6, 1, 5, 11, 10, 1, 6, 18, 26, 15, 1, 7, 27, 58, 57, 21, 1, 8, 38, 112, 179, 120, 28, 1, 9, 51, 194, 453, 543, 247, 36

Offset: 0

Gary W. Adamson, May 30 2005

By diagonals (d=1,2,3,...) going to the left with (1,3,6,...) = d(1), these are sequences of the form (k-th term a(k) = d*a(k-1) + k). Example: 1, 7, 38, 194, ... (the 5th diagonal) = A014827, is generated by a(k) = 5*a(k-1) + k. Diagonal 2 = (1, 4, 11, 26, ...) = A000295; Diagonal 3 = (1, 5, 18, ...) = A000340; Diagonal 4 = (1, 6, 27, ...) = A014825.

Triangle A108243 is generated by analogous operations from (..., 3, 2, 1) instead of (1, 2, 3, ...).

			4th column (offset) = 10, 26, 58, 112, ...= f(x), x = 1, 2, 3; x^3 + 2x^2 + 3x + 4.
First few rows of the triangle are:
  1;
  1, 3;
  1, 4, 6;
  1, 5, 11, 10;
  1, 6, 18, 26, 15;
  1, 7, 27, 58, 57, 21;
  1, 8, 38, 112, 179, 120, 28;
  ...

Cf. A108286, A000295, A000349, A014825, A000217, A108283, A108284, A108286.

n-th column = f(x), x = 1, 2, 3, ...; x^(n) + 2*x^(n-1) + 3*x^(n-2) + ... + (n+1).

A276659 Accumulation of the upper left triangle used in binomial transform of nonnegative integers.

Original entry on oeis.org

0, 2, 11, 39, 114, 300, 741, 1757, 4052, 9162, 20415, 44979, 98214, 212888, 458633, 982905, 2097000, 4456278, 9436995, 19922735, 41942810, 88080132, 184549101, 385875669, 805306044, 1677721250, 3489660551, 7247756907, 15032385102, 31138512432, 64424508945

Offset: 0

Jean-François Alcover and Francois Alcover, Sep 11 2016

After 0, is this the second column of A108284? [Bruno Berselli, Sep 13 2016 - this comment may be removed if the property is confirmed.]

			Starting from the triangle:
   0,  1,  2,  3,  4,  5, ...
   1,  3,  5,  7,  9, ...
   4,  8, 12, 16, ...
  12, 20, 28, ...
  32, 48, ...
  80, ...
  ...
the first terms are:
a(0) = 0;
a(1) = a(0) + 1 + 1 = 2;
a(2) = a(1) + 4 + 3 + 2 = 11;
a(3) = a(2) + 12 + 8 + 5 + 3 = 39, etc.
First column is A001787: n*2^(n-1).

Index entries for linear recurrences with constant coefficients, signature (7,-19,25,-16,4).

Cf. A001787, A058877, A062111, A152920.

[(2^(n+2)-n-3)*n/2: n in [0..40]]; // Vincenzo Librandi, Sep 13 2016

A276659:=n->n*(2^(n+2) - n - 3)/2: seq(A276659(n), n=0..50); # Wesley Ivan Hurt, Sep 16 2017

t[0, k_] := k; t[n_, k_] := t[n, k] = t[n - 1, k] + t[n - 1, k + 1]; a[n_] := Sum[t[m, k], {m, 0, n}, {k, 0, n - m}]; Table[a[n], {n, 0, 30}]
Table[(2^(n + 2) - n - 3) n / 2, {n, 0, 30}] (* Vincenzo Librandi, Sep 13 2016 *)

x='x+O('x^99); concat(0, Vec(x*(2-3*x)/((1-x)^3*(1-2*x)^2))) \\ Altug Alkan, Sep 14 2017

O.g.f.: x*(2 - 3*x)/((1 - x)^3*(1 - 2*x)^2).
E.g.f.: x*exp(x)*(8*exp(x) - x - 4)/2.
a(n) = n*(2^(n+2) - n - 3)/2.
a(n) = 7*a(n-1) - 19*a(n-2) + 25*a(n-3) - 16*a(n-4) + 4*a(n-5) for n > 4.
a(n) = a(n-1) + A058877(n+1). - R. J. Mathar, Sep 14 2016
a(n) = Sum_{k=2..n+3} Sum_{i=2..n+3} k * C(n-i+3,k). - Wesley Ivan Hurt, Sep 20 2017

Edited and extended by Bruno Berselli, Sep 13 2016

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A108283 Triangle read by rows, generated from (..., 3, 2, 1).

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A108285 Triangle read by rows, generated from (1, 2, 3, ...).

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A276659 Accumulation of the upper left triangle used in binomial transform of nonnegative integers.

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