A213569 -id:A213569 - cp's OEIS frontend

A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

1, 4, 2, 10, 7, 3, 20, 16, 10, 4, 35, 30, 22, 13, 5, 56, 50, 40, 28, 16, 6, 84, 77, 65, 50, 34, 19, 7, 120, 112, 98, 80, 60, 40, 22, 8, 165, 156, 140, 119, 95, 70, 46, 25, 9, 220, 210, 192, 168, 140, 110, 80, 52, 28, 10, 286, 275, 255, 228, 196, 161, 125, 90

Offset: 1

Clark Kimberling, Jun 14 2012

Principal diagonal: A002412.
Antidiagonal sums: A002415.
Row 1:  (1,2,3,...)**(1,2,3,...) = A000292.
Row 2:  (1,2,3,...)**(2,3,4,...) = A005581.
Row 3:  (1,2,3,...)**(3,4,5,...) = A006503.
Row 4:  (1,2,3,...)**(4,5,6,...) = A060488.
Row 5:  (1,2,3,...)**(5,6,7,...) = A096941.
Row 6:  (1,2,3,...)**(6,7,8,...) = A096957.
...
In general, the convolution of two infinite sequences is defined from the convolution of two n-tuples:  let X(n) = (x(1),...,x(n)) and Y(n)=(y(1),...,y(n)); then X(n)**Y(n) = x(1)*y(n)+x(2)*y(n-1)+...+x(n)*y(1); this sum is the n-th term in the convolution of infinite sequences:(x(1),...,x(n),...)**(y(1),...,y(n),...), for all n>=1.
...
In the following guide to related arrays and sequences, row n of each array T(n,k) is the convolution b**c of the sequences b(h) and c(h+n-1). The principal diagonal is given by T(n,n) and the n-th antidiagonal sum by S(n). In some cases, T(n,n) or S(n) differs in offset from the listed sequence.
b(h)........ c(h)........ T(n,k) .. T(n,n) .. S(n)
h .......... h .......... A213500 . A002412 . A002415
h .......... h^2 ........ A212891 . A213436 . A024166
h^2 ........ h .......... A213503 . A117066 . A033455
h^2 ........ h^2 ........ A213505 . A213546 . A213547
h .......... h*(h+1)/2 .. A213548 . A213549 . A051836
h*(h+1)/2 .. h .......... A213550 . A002418 . A005585
h*(h+1)/2 .. h*(h+1)/2 .. A213551 . A213552 . A051923
h .......... h^3 ........ A213553 . A213554 . A101089
h^3 ........ h .......... A213555 . A213556 . A213547
h^3 ........ h^3 ........ A213558 . A213559 . A213560
h^2 ........ h*(h+1)/2 .. A213561 . A213562 . A213563
h*(h+1)/2 .. h^2 ........ A213564 . A213565 . A101094
2^(h-1) .... h .......... A213568 . A213569 . A047520
2^(h-1) .... h^2 ........ A213573 . A213574 . A213575
h .......... Fibo(h) .... A213576 . A213577 . A213578
Fibo(h) .... h .......... A213579 . A213580 . A053808
Fibo(h) .... Fibo(h) .... A067418 . A027991 . A067988
Fibo(h+1) .. h .......... A213584 . A213585 . A213586
Fibo(n+1) .. Fibo(h+1) .. A213587 . A213588 . A213589
h^2 ........ Fibo(h) .... A213590 . A213504 . A213557
Fibo(h) .... h^2 ........ A213566 . A213567 . A213570
h .......... -1+2^h ..... A213571 . A213572 . A213581
-1+2^h ..... h .......... A213582 . A213583 . A156928
-1+2^h ..... -1+2^h ..... A213747 . A213748 . A213749
h .......... 2*h-1 ...... A213750 . A007585 . A002417
2*h-1 ...... h .......... A213751 . A051662 . A006325
2*h-1 ...... 2*h-1 ...... A213752 . A100157 . A071238
2*h-1 ...... -1+2^h ..... A213753 . A213754 . A213755
-1+2^h ..... 2*h-1 ...... A213756 . A213757 . A213758
2^(n-1) .... 2*h-1 ...... A213762 . A213763 . A213764
2*h-1 ...... Fibo(h) .... A213765 . A213766 . A213767
Fibo(h) .... 2*h-1 ...... A213768 . A213769 . A213770
Fibo(h+1) .. 2*h-1 ...... A213774 . A213775 . A213776
Fibo(h) .... Fibo(h+1) .. A213777 . A001870 . A152881
h .......... 1+[h/2] .... A213778 . A213779 . A213780
1+[h/2] .... h .......... A213781 . A213782 . A005712
1+[h/2] .... [(h+1)/2] .. A213783 . A213759 . A213760
h .......... 3*h-2 ...... A213761 . A172073 . A002419
3*h-2 ...... h .......... A213771 . A213772 . A132117
3*h-2 ...... 3*h-2 ...... A213773 . A214092 . A213818
h .......... 3*h-1 ...... A213819 . A213820 . A153978
3*h-1 ...... h .......... A213821 . A033431 . A176060
3*h-1 ...... 3*h-1 ...... A213822 . A213823 . A213824
3*h-1 ...... 3*h-2 ...... A213825 . A213826 . A213827
3*h-2 ...... 3*h-1 ...... A213828 . A213829 . A213830
2*h-1 ...... 3*h-2 ...... A213831 . A213832 . A212560
3*h-2 ...... 2*h-1 ...... A213833 . A130748 . A213834
h .......... 4*h-3 ...... A213835 . A172078 . A051797
4*h-3 ...... h .......... A213836 . A213837 . A071238
4*h-3 ...... 2*h-1 ...... A213838 . A213839 . A213840
2*h-1 ...... 4*h-3 ...... A213841 . A213842 . A213843
2*h-1 ...... 4*h-1 ...... A213844 . A213845 . A213846
4*h-1 ...... 2*h-1 ...... A213847 . A213848 . A180324
[(h+1)/2] .. [(h+1)/2] .. A213849 . A049778 . A213850
h .......... C(2*h-2,h-1) A213853
...
Suppose that u = (u(n)) and v = (v(n)) are sequences having generating functions U(x) and V(x), respectively. Then the convolution u**v has generating function U(x)*V(x). Accordingly, if u and v are homogeneous linear recurrence sequences, then every row of the convolution array T satisfies the same homogeneous linear recurrence equation, which can be easily obtained from the denominator of U(x)*V(x). Also, every column of T has the same homogeneous linear recurrence as v.

			Northwest corner (the array is read by southwest falling antidiagonals):
  1,  4, 10, 20,  35,  56,  84, ...
  2,  7, 16, 30,  50,  77, 112, ...
  3, 10, 22, 40,  65,  98, 140, ...
  4, 13, 28, 50,  80, 119, 168, ...
  5, 16, 34, 60,  95, 140, 196, ...
  6, 19, 40, 70, 110, 161, 224, ...
T(6,1) = (1)**(6) = 6;
T(6,2) = (1,2)**(6,7) = 1*7+2*6 = 19;
T(6,3) = (1,2,3)**(6,7,8) = 1*8+2*7+3*6 = 40.

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A000027.

b[n_] := n; c[n_] := n
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213500 *)

t(n,k) = sum(i=0, k - 1, (k - i) * (n + i));
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(t(k,n - k + 1),", ");); print(););};
tabl(12) \\ Indranil Ghosh, Mar 26 2017

def t(n, k): return sum((k - i) * (n + i) for i in range(k))
for n in range(1, 13):
    print([t(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 26 2017

T(n,k) = 4*T(n,k-1) - 6*T(n,k-2) + 4*T(n,k-3) - T(n,k-4).
T(n,k) = 2*T(n-1,k) - T(n-2,k).
G.f. for row n:  x*(n - (n - 1)*x)/(1 - x)^4.

A213568 Rectangular array: (row n) = bc, where b(h) = 2^(h-1), c(h) = n-1+h, n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 4, 2, 11, 7, 3, 26, 18, 10, 4, 57, 41, 25, 13, 5, 120, 88, 56, 32, 16, 6, 247, 183, 119, 71, 39, 19, 7, 502, 374, 246, 150, 86, 46, 22, 8, 1013, 757, 501, 309, 181, 101, 53, 25, 9, 2036, 1524, 1012, 628, 372, 212, 116, 60, 28, 10, 4083, 3059, 2035, 1267

Offset: 1

Clark Kimberling, Jun 18 2012

Principal diagonal:  A213569
Antidiagonal sums:  A047520
Row 1,  (1,3,6,...)**(1,4,9,...):  A125128
Row 2,  (1,3,6,...)**(4,9,16,...):  A095151
Row 3,  (1,3,6,...)**(9,16,25,...):  A000247
Row 4,  (1,3,6,...)**(16,25,36...):  A208638 (?)
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
  1...4....11...26....57....120
  2...7....18...41....88....183
  3...10...25...56....119...246
  4...13...32...71....150...309
  5...16...39...86....181...372
  6...19...46...101...212...435

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A213500.

Flat(List([1..12], n-> List([1..n], k-> 2^(n-k+1)*(k+1) -(n+2) ))); # G. C. Greubel, Jul 26 2019

[2^(n-k+1)*(k+1) -(n+2): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 26 2019

(* First program *)
b[n_]:= 2^(n-1); c[n_]:= n;
t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213568 *)
d = Table[t[n, n], {n, 1, 40}] (* A213569 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A047520 *)
(* Second program *)
Table[2^(n-k+1)*(k+1) -(n+2), {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 26 2019 *)

for(n=1,12, for(k=1,n, print1(2^(n-k+1)*(k+1) -(n+2), ", "))) \\ G. C. Greubel, Jul 26 2019

[[2^(n-k+1)*(k+1) -(n+2) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 26 2019

T(n,k) = 4*T(n,k-1) - 5*T(n,k-2) + 2*T(n,k-3). - corrected by Clark Kimberling, Sep 03 2023
G.f. for row n:  f(x)/g(x), where f(x) = n - (n - 1)*x and g(x) = (1 - 2*x)*(1 - x)^2.
T(n,k) = 2^k*(n + 1) - (n + k + 1). - G. C. Greubel, Jul 26 2019

A213574 Principal diagonal of the convolution array A213573.

Original entry on oeis.org

1, 17, 93, 349, 1093, 3093, 8221, 20957, 51861, 125509, 298477, 699789, 1621285, 3718325, 8453181, 19069885, 42728245, 95156901, 210762253, 464517485, 1019214021, 2227173397, 4848613213, 10519312029, 22749902293, 49056576773, 105495131181, 226291086157

Offset: 1

Clark Kimberling, Jun 18 2012

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-33,63,-66,36,-8).

Cf. A213568, A213500.

List([1..30], n-> 2^n*(3+2*n+n^2) - (3+4*n+4*n^2)); # G. C. Greubel, Jul 25 2019

[2^n*(3+2*n+n^2) - (3+4*n+4*n^2): n in [1..30]]; // G. C. Greubel, Jul 25 2019

(* First program *)
b[n_]:= 2^(n-1); c[n_]:= n;
t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213568 *)
d = Table[t[n, n], {n, 1, 40}] (* A213569 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A047520 *)
(* Additional programs *)
LinearRecurrence[{9,-33,63,-66,36,-8},{1,17,93,349,1093,3093},30] (* Harvey P. Dale, Jun 25 2014 *)
Rest[CoefficientList[Series[x(1+8x-27x^2+10x^3+16x^4)/(1-3x+2x^2)^3, {x, 0, 30}], x]] (* Vincenzo Librandi, Jun 26 2014 *)

Vec(x*(1+8*x-27*x^2+10*x^3+16*x^4)/((1-x)^3*(1-2*x)^3) + O(x^30)) \\ Colin Barker, Oct 30 2017

vector(30, n, 2^n*(3+2*n+n^2) - (3+4*n+4*n^2)) \\ G. C. Greubel, Jul 25 2019

[2^n*(3+2*n+n^2) - (3+4*n+4*n^2) for n in (1..30)] # G. C. Greubel, Jul 25 2019

a(n) = 9*a(n-1) - 33*a(n-2) + 63*a(n-3) - 66*a(n-4) + 36*a(n-5) - 8*a(n-6).
G.f.: x*(1 + 8*x - 27*x^2 + 10*x^3 + 16*x^4)/(1 - 3*x + 2*x^2)^3.
a(n) = 2^n*(3+2*n+n^2) - (3+4*n+4*n^2). - Colin Barker, Oct 30 2017
E.g.f.: (3+6*x+4*x^2)*exp(2*x) - (3+8*x+4*x^2)*exp(x). - G. C. Greubel, Jul 25 2019

A327916 Triangle T(k, n) read by rows: Array A(k, n) = 2^k(k + 1 + 2n), k >= 0, n >= 0, read by antidiagonals upwards.

Original entry on oeis.org

1, 4, 3, 12, 8, 5, 32, 20, 12, 7, 80, 48, 28, 16, 9, 192, 112, 64, 36, 20, 11, 448, 256, 144, 80, 44, 24, 13, 1024, 576, 320, 176, 96, 52, 28, 15, 2304, 1280, 704, 384, 208, 112, 60, 32, 17, 5120, 2816, 1536, 832, 448, 240, 128, 68, 36, 19, 11264, 6144, 3328, 1792, 960, 512, 272, 144, 76, 40, 21

Offset: 0

Wolfdieter Lang, Oct 03 2019

The array A(k, n) arises from the following Pascal-type triangles PTodd(k), k >= 0 based on the positive odd integers A005408.
For example, the Pascal-type triangle PTodd(k), for k = 3 is
   1     3     5     7
      4     8    12
        12    20
           32
Taken upside-down such triangles become so-called addition towers of height k+1 (Rechenturm in German elementary schools; thanks to my correspondent Bennet D.), starting with any k+1 numbers. Here the positive odd numbers are used.
The sequence s of the final number of these Pascal-type triangles PT(k), for k >= 0, begins 1, 4, 12, 32, ...; s(k) = (k+1)*2^k = A001787(k+1), for k >= 0.
For k -> infinity the left-aligned row sequences build the array A(k, n), with k >= 0 and n >= 0, namely A(k, n) = 2^k*(k + 2*n + 1); this array begins:
  k\n  0   1   2   3   4    5 ...
  -------------------------------
  0:   1   3   5   7   9   11 ... {A005408(n)}
  1:   4   8  12  16  20   24 ... {A008586(n+1)}
  2:  12  20  28  36  44   52 ... {A017113(n+1)}
  3:  32  48  64  80  96  112 ... {A008598(n+2)}
  4:  80 112 144 176 208  240 ... {16*A005408(n+2)}
  5: 192 256 320 384 448  512 ... {A152691(n+3)}
  6: 448 576 704 832 960 1088 ... {64*A005408(n+3)}
  ...
The sequence s, the first (n=0) column of A, is always the binomial transform of the first (k=0) row in A.
A(k, n) = Sum_{j=0..k} binomial(k, j)*(2*(n+j)+1) = 2^k*(k + 1 + 2*n), for k >= 0 and n >= 0.
The corresponding antidiagonal-upwards read triangle is T(k, n) = A(k-n, n) = 2^(k-n)*(k + n + 1), n >= 0, k = 0..n.
If the nonnegative integers A001477 are used as k = 0 row of the array Anneg(k, n) = 2^(k-1)*(2*n + k), for k >= 0, n >= 0, with the triangle Tnneg(k, n) = Anneg(k-n, n) = (n + k)*2^(k-n-1), k >= 0, n = 0..k, then the s sequence is snneg(k) = Tnneg(k, 0) = k*2^{k-1} = A001787(k), the binomial transform of the sequence{A001477(n)}_{n>=0}. The triangle Tnneg begins [0], [1, 1], [4, 3, 2], [12, 8, 5, 3], [32, 20, 12, 7, 4], ... . See A062111 and the row-reversed triangle A152920 for other versions.

			The triangle T(k, n) begins:
   k\n    0    1    2    3   4   5   6   7  8  9 10 ...
  -----------------------------------------------------
   0:     1
   1:     4    3
   2:    12    8    5
   3:    32   20   12    7
   4:    80   48   28   16   9
   5:   192  112   64   36  20  11
   6:   448  256  144   80  44  24  13
   7:  1024  576  320  176  96  52  28  15
   8:  2304 1280  704  384 208 112  60  32 17
   9:  5120 2816 1536  832 448 240 128  68 36 19
  10: 11264 6144 3328 1792 960 512 272 144 76 40 21
  ...

Column sequences without leading zeros are for n=0..9: A001787(n+1), A001792(n+1), A045623(n+2), A045891(n+3), A034007(n+4), A111297(n+3), A159694(n+1), A159695(n+1), A159696(n+1), A159697(n+1).
The sequence of (sub)diagonal k, for k >= 0, is the row k sequence of array A: {(k + 2*n + 1)*2^k}_{k >= 0}.
Row sums: A213569(k+1), k >= 0 (see the J. M. Bergot comments there).
Cf. A006211, A152920.

Table[2^#*(# + 1 + 2 n) &[k - n], {k, 0, 10}, {n, 0, k}] // Flatten (* Michael De Vlieger, Oct 03 2019 *)

Array A(k, n) = Sum_{j=0..k} binomial(k, j)*(2*(n+j) + 1) = 2^k*(k + 1+ 2*n), for k >= 0 and n >= 0.
Triangle T(k, n) =  A(k-n, n) = 2^(k-n)*(k + n + 1), n >= 0, k = 0..n.
Recurrence: T(k, 0) = (k+1)*2^k = A001787(k+1), for k >= 0, and  T(k, n) = T(k, n-1) - T(k-1, n-1), for n >= 1, k >= 1, with T(k, n) = 0 if k < n.
O.g.f. for row polynomials: G(z,x) = Sum_{n=0..k} R(k, x)*z^n =
(1  + x*z*(1 - 4*z))/((1 - 2*z)^2*(1 - x*z)^2).
T(k, 0) = Sum_{n=0..k} binomial(k,n)*T(n, n), k >= 0 (binomial transform).

Definition corrected by Georg Fischer, Jul 13 2023

cp's OEIS Frontend

A213500 Rectangular array T(n,k): (row n) = b**c, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and ** = convolution.

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A213568 Rectangular array: (row n) = b**c, where b(h) = 2^(h-1), c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

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A213574 Principal diagonal of the convolution array A213573.

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A327916 Triangle T(k, n) read by rows: Array A(k, n) = 2^k*(k + 1 + 2*n), k >= 0, n >= 0, read by antidiagonals upwards.

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A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

A213568 Rectangular array: (row n) = bc, where b(h) = 2^(h-1), c(h) = n-1+h, n>=1, h>=1, and = convolution.

A327916 Triangle T(k, n) read by rows: Array A(k, n) = 2^k(k + 1 + 2n), k >= 0, n >= 0, read by antidiagonals upwards.