A213500 -id:A213500 - cp's OEIS frontend

A213564 Rectangular array: (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.

1, 7, 4, 27, 21, 9, 77, 67, 43, 16, 182, 167, 127, 73, 25, 378, 357, 297, 207, 111, 36, 714, 686, 602, 467, 307, 157, 49, 1254, 1218, 1106, 917, 677, 427, 211, 64, 2079, 2034, 1890, 1638, 1302, 927, 567, 273, 81, 3289, 3234, 3054, 2730, 2282, 1757

Offset: 1

Clark Kimberling, Jun 18 2012

Principal diagonal:  A213565
Antidiagonal sums:  A101094
Row 1,  (1,3,6,...)**(1,4,9,...):  A005585
Row 2,  (1,3,6,...)**(4,9,16,...):  (k^5 +25*k^4 + 60*k^3 + 215*k^2 + 59*k)/60
Row 3,  (1,3,6,...)**(9,16,25,...):  (k^5 +35*k^4 + 30*k^3 + 505*k^2 + 149*k)/60
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....7.....27....77....182
4....21....67....167...357
9....43....127...297...602
16...73....207...467...917
25...111...307...677...1302
36...157...427...927...1757

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

Cf. A213500.

b[n_] := n (n + 1)/2; c[n_] := n^2
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}]  (* A213564 *)
d = Table[t[n, n], {n, 1, 40}] (* A213565 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A101094 *)

T(n,k) = 6*T(n,k-1) - 15*T(n,k-2) + 20*T(n,k-3) - 15*T(n,k-4) + 6*T(n,k-5) - T(n,k-6).

G.f. for row n: f(x)/g(x), where f(x) = n^2 - (2*n^2 - 2n - 1)*x + ((n - 1)^2)*x^2 and g(x) = (1 - x)^6.

A213569 Principal diagonal of the convolution array A213568.

Original entry on oeis.org

1, 7, 25, 71, 181, 435, 1009, 2287, 5101, 11243, 24553, 53223, 114661, 245731, 524257, 1114079, 2359261, 4980699, 10485721, 22020055, 46137301, 96468947, 201326545, 419430351, 872415181, 1811939275, 3758096329, 7784628167

Offset: 1

Clark Kimberling, Jun 18 2012

Create a triangle having first column T(n,1) = 2*n-1 for n = 1,2,3... The remaining terms are set to T(r,c) = T(r,c-1) + T(r-1,c-1). The sum of the terms in row n is a(n). The first five rows of the triangle are 1; 3,4; 5,8,12; 7,12,20,32; 9,16,28,48,80. - J. M. Bergot, Jan 17 2013

Starting at n=1, a(n) = (n+1)*2^n - 2*n - 1. A001787(n) = n*2^n. - J. M. Bergot, Jan 27 2013

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A001787, A213568, A213500.

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

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

f:= gfun:-rectoproc({a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4),
  a(1)=1,a(2)=7,a(3)=25,a(4)=71},a(n),remember):
map(f, [$1..30]); # Robert Israel, Sep 19 2017

(* 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[{6,-13,12,-4},{1,7,25,71},30] (* Harvey P. Dale, Jan 06 2015 *)
Table[2^n*(n+1) -(2*n+1), {n,30}] (* G. C. Greubel, Jul 25 2019 *)

my(x='x+O('x^30)); Vec(x*(1+x-4*x^2)/((1-2*x)^2*(1-x)^2)) \\ Altug Alkan, Sep 19 2017

vector(30, n, 2^n*(n+1) -(2*n+1)) \\ G. C. Greubel, Jul 25 2019

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

a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4).
G.f.: x*(1 + x - 4*x^2)/( (1-2*x)^2*(1-x)^2 ).
a(n) = A001787(n+1)- 2*n - 1. - J. M. Bergot, Jan 22 2013
a(n) = Sum_{k=1..n} Sum_{i=0..n} (n-i) * C(k,i). - Wesley Ivan Hurt, Sep 19 2017

A213571 Rectangular array: (row n) = bc, where b(h) = h, c(h) = (n-1+h)^2, n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 5, 3, 16, 13, 7, 42, 38, 29, 15, 99, 94, 82, 61, 31, 219, 213, 198, 170, 125, 63, 466, 459, 441, 406, 346, 253, 127, 968, 960, 939, 897, 822, 698, 509, 255, 1981, 1972, 1948, 1899, 1809, 1654, 1402, 1021, 511, 4017, 4007, 3980, 3924, 3819, 3633

Offset: 1

Clark Kimberling, Jun 19 2012

Principal diagonal:  A213572.
Antidiagonal sums:  A213581.
Row 1,  (1,2,3,4,5,...)**(1,3,7,15,31,...): A002662.
Row 2,  (1,2,3,4,5,...)**(3,7,15,31,63,...).
Row 3,  (1,2,3,4,5,...)**(7,15,31,63,...).
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
   1,    5,   16,   42,   99,  219, ...
   3,   13,   38,   94,  213,  459, ...
   7,   29,   82,  198,  441,  939, ...
  15,   61,  170,  406,  897, 1899, ...
  31,  125,  346,  822, 1809, 3819, ...
  ...

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

Cf. A213500, A213587.

Flat(List([1..12], n-> List([1..n], k-> 2^(n+2) -2^k*(n-k+3) -Binomial(n-k+2, 2) ))); # G. C. Greubel, Jul 25 2019

[2^(n+2) -2^k*(n-k+3) -Binomial(n-k+2, 2): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 25 2019

(* First program *)
b[n_]:= n; c[n_]:= -1 + 2^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}]  (* A213571 *)
d = Table[t[n, n], {n, 1, 40}] (* A213572 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213581 *)
(* Additional programs *)
Table[2^(n+2) -2^k*(n-k+3) -Binomial[n-k+2, 2], {n,12}, {k, n}]//Flatten (* G. C. Greubel, Jul 25 2019 *)

for(n=1,12, for(k=1,n, print1(2^(n+2) -2^k*(n-k+3) -binomial(n-k+2, 2), ", "))) \\ G. C. Greubel, Jul 25 2019

[[2^(n+2) -2^k*(n-k+3) -binomial(n-k+2, 2) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 25 2019

T(n,k) = 5*T(n,k-1) - 9*T(n,k-2) + 7*T(n,k-3) - 2*T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = x*(-1 + 2^n - (-2 + 2^n)*x) and g(x) = (1 - 2*x)(1 - x)^3.
T(n,k) = 2^(n+k+1) - 2^n*(k+2) - binomial(k+1, 2). - G. C. Greubel, Jul 25 2019

A213584 Rectangular array: (row n) = bc, where b(h) = F(h+1), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 4, 2, 10, 7, 3, 21, 16, 10, 4, 40, 32, 22, 13, 5, 72, 59, 43, 28, 16, 6, 125, 104, 78, 54, 34, 19, 7, 212, 178, 136, 97, 65, 40, 22, 8, 354, 299, 231, 168, 116, 76, 46, 25, 9, 585, 496, 386, 284, 200, 135, 87, 52, 28, 10, 960, 816, 638, 473, 337, 232, 154, 98, 58, 31, 11

Offset: 1

Clark Kimberling, Jun 18 2012

Principal diagonal: A213585.
Antidiagonal sums: A213586.
Row 1, (1,2,3,5,...)**(1,2,3,4,...): A001891.
Row 2, (1,2,3,5,...)**(2,3,4,5,...): A023550.
Row 3, (1,2,3,5,...)**(3,4,5,6,...): A023554.
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1...4....10...21...40....72
2...7....16...32...59....104
3...10...22...43...78....136
4...13...28...54...97....168
5...16...34...65...116...200
6...19...40...76...135...232

Clark Kimberling, Antidiagonals n = 1..60

Cf. A000045, A213500, A213579.

Flat(List([1..12], n-> List([1..n], k-> Fibonacci(n-k+5) + k*Fibonacci(n-k+4) -(2*n+5)))) # G. C. Greubel, Jul 08 2019

[[Fibonacci(n-k+5) + k*Fibonacci(n-k+4) -(2*n+5): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 08 2019

(* First program *)
b[n_]:= Fibonacci[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}]] (* A213584 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
d = Table[T[n, n], {n, 1, 40}] (* A213585 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213586 *)
(* Second program *)
Table[Fibonacci[n-k+5] + k*Fibonacci[n-k+4] -2*n-5, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 08 2019 *)

t(n,k) = fibonacci(n-k+5) + k*fibonacci(n-k+4) -(2*n+5);
for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 08 2019

[[fibonacci(n-k+5) + k*fibonacci(n-k+4) -(2*n+5) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 08 2019

T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - T(n,k-3) + T(n,k-4).
G.f. for row n:  f(x)/g(x), where f(x) = n + x - (n - 1)*x and g(x) = (1 - x - x^2)*(1 - x)^2.
T(n, k) = Fibonacci(k+4) + n*Fibonacci(k+3) - 2*(n+k) - 3. - G. C. Greubel, Jul 08 2019

A213586 Antidiagonal sums of the convolution array A213584.

Original entry on oeis.org

1, 6, 20, 51, 112, 224, 421, 758, 1324, 2263, 3808, 6336, 10457, 17158, 28036, 45675, 74256, 120544, 195485, 316790, 513116, 830831, 1344960, 2176896, 3523057, 5701254, 9225716, 14928483, 24155824, 39086048, 63243733, 102331766

Offset: 1

Clark Kimberling, Jun 18 2012

a(n) is the number of bit strings of length n+5 with the pattern 01 at least thrice, and without the pattern 110, see example. - John M. Campbell, Jan 25 2013

			From _John M. Campbell_, Jan 25 2013: (Start)
There are a(3) = 20 bit strings of length 3+5 with the pattern 01 at least thrice, and without the pattern 110:
00010101, 00100101, 00101001, 00101010, 00101011,
01000101, 01001001, 01001010, 01001011, 01010001,
01010010, 01010011, 01010100, 01010101, 01010111,
10010101, 10100101, 10101001, 10101010, 10101011.
(End)

Clark Kimberling, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Cf. A213500, A213584, A213590.

List([1..40], n-> Fibonacci(n+8) -(21+10*n+2*n^2)) # G. C. Greubel, Jul 06 2019

[Fibonacci(n+8) -(21+10*n+2*n^2): n in [1..40]]; // G. C. Greubel, Jul 06 2019

(See A213584.)
With[{F = Fibonacci}, Table[F[n+8] -(21+10*n+2*n^2), {n,40}]] (* G. C. Greubel, Jul 06 2019 *)

vector(40, n, fibonacci(n+8) -(21+10*n+2*n^2)) \\ G. C. Greubel, Jul 06 2019

[fibonacci(n+8) -(21+10*n+2*n^2) for n in (1..40)] # G. C. Greubel, Jul 06 2019

a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
G.f.: x*(1 + 2*x + x^2)/((1 - x - x^2)*(1 - x)^3).
a(n) = Fibonacci(n+8) - (21 + 10*n + 2*n^2). - G. C. Greubel, Jul 06 2019

A213750 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 2*(n-1+h)-1, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 5, 3, 14, 11, 5, 30, 26, 17, 7, 55, 50, 38, 23, 9, 91, 85, 70, 50, 29, 11, 140, 133, 115, 90, 62, 35, 13, 204, 196, 175, 145, 110, 74, 41, 15, 285, 276, 252, 217, 175, 130, 86, 47, 17, 385, 375, 348, 308, 259, 205, 150, 98, 53, 19, 506, 495, 465, 420

Offset: 1

Clark Kimberling, Jun 20 2012

Principal diagonal:  A007585
Antidiagonal sums:  A002417
row 1,  (1,2,3,4,5,...)**(1,3,5,7,9,...): A000330
row 2,  (1,2,3,4,5,...)**(3,5,7,9,...): A051925
row 3,  (1,2,3,4,5,...)**(5,7,9,11,...): (2*k^3 + 15*k^2 + 13*k)/6
row 4,  (1,2,3,4,5,...)**(7,9,11,13,...): (2*k^3 + 21*k^2 + 19*k)/6
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....5....14...30....55....91
3....11...26...50....85....133
5....17...38...70....115...175
7....23...50...90....145...217
9....29...62...110...175...259
11...35...74...130...205...301

Cf. A213500.

b[n_] := n; c[n_] := 2 n - 1;
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}]  (* A213750 *)
d = Table[t[n, n], {n, 1, 40}] (* A007585 *)
s1 = Table[s[n], {n, 1, 50}] (* A002417 *)
FindLinearRecurrence[s1]
FindGeneratingFunction[s1, x]

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = (2*n - 1) - (2*n - 3)*x and g(x) = (1 - x )^4.

A213751 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 5, 2, 14, 9, 3, 30, 23, 13, 4, 55, 46, 32, 17, 5, 91, 80, 62, 41, 21, 6, 140, 127, 105, 78, 50, 25, 7, 204, 189, 163, 130, 94, 59, 29, 8, 285, 268, 238, 199, 155, 110, 68, 33, 9, 385, 366, 332, 287, 235, 180, 126, 77, 37, 10, 506, 485, 447, 396, 336, 271

Offset: 1

Clark Kimberling, Jun 20 2012

Principal diagonal:  A051662
Antidiagonal sums:  A006325
row 1,  (1,3,5,7,9,...)**(1,2,3,4,5,...): A000330
row 2,  (1,3,5,7,9,...)**(2,3,4,5,6,...): A101986
row 3,  (1,3,5,7,9,...)**(3,4,5,6,7,...): (2*k^3 + 15*k^2 + k)/6
row 4,  (1,3,5,7,9,...)**(4,5,6,7,8,...): (2*k^3 + 21*k^2 + k)/6
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1...5....14...30....55....91
2...9....23...46....80....127
3...13...32...62....105...163
4...17...41...78....130...199
5...21...50...94....155...235
6...25...59...110...180...271

Cf. A213500.

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}]  (* A213751 *)
Table[t[n, n], {n, 1, 40}] (* A051662 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A006325 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = n + x - (n - 1)*x^2 and g(x) = (1 - x )^4.

A213752 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = b(n-1+h), n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 6, 3, 19, 14, 5, 44, 37, 22, 7, 85, 76, 55, 30, 9, 146, 135, 108, 73, 38, 11, 231, 218, 185, 140, 91, 46, 13, 344, 329, 290, 235, 172, 109, 54, 15, 489, 472, 427, 362, 285, 204, 127, 62, 17, 670, 651, 600, 525, 434, 335, 236, 145, 70, 19, 891, 870, 813

Offset: 1

Clark Kimberling, Jun 20 2012

Principal diagonal:  A100157
Antidiagonal sums:  A071238
row 1,  (1,3,5,7,9,...)**(1,3,5,7,9,...): A005900
row 2,  (1,3,5,7,9,...)**(3,5,7,9,11,...): A143941
row 3,  (1,3,5,7,9,...)**(5,7,9,11,13,...): (2*k^3 + 12*k^2 + k)/6
row 4,  (1,3,5,7,9,...)**(7,9,11,13,15,,...): (2*k^3 + 18*k^2 + k)/6
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1...6....19...44....85....146
3...14...37...76....135...218
5...22...55...108...185...290
7...30...73...140...235...362
9...38...91...172...285...434

Cf. A213500.

b[n_] := 2 n - 1; c[n_] := 2 n - 1;
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}]  (* A213752 *)
Table[t[n, n], {n, 1, 40}] (* A100157 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A071238 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = 2*n - 1 + 2*x - (2*n - 3)*x^2 and g(x) = (1 - x )^4.

A213756 Rectangular array: (row n) = b**c, where b(h) = -1 + 2^h, c(h) = 2n - 3 + 2h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 6, 3, 21, 14, 5, 58, 43, 22, 7, 141, 110, 65, 30, 9, 318, 255, 162, 87, 38, 11, 685, 558, 369, 214, 109, 46, 13, 1434, 1179, 798, 483, 266, 131, 54, 15, 2949, 2438, 1673, 1038, 597, 318, 153, 62, 17, 5998, 4975, 3442, 2167, 1278, 711, 370, 175, 70

Offset: 1

Clark Kimberling, Jun 20 2012

Principal diagonal: A213757.
Antidiagonal sums: A213758.
Row 1,  (1,3,7,15,31,...)**(1,3,5,7,9,...): A047520.
Row 2,  (1,3,7,15,31,...)**(3,5,7,9,11,...).
Row 3,  (1,3,7,15,31,...)**(5,7,9,11,13,...).
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....6....21....58....141...318
3....14...43....110...255...558
5....22...65....162...369...798
7....30...87....214...483...1038
9....38...109...266...597...1278
11...46...131...318...711...1518

Clark Kimberling, Antidiagonals n = 1..40, flattened

Cf. A213500.

b[n_] := -1 + 2^n; c[n_] := 2 n - 1;
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}]  (* A213756 *)
Table[t[n, n], {n, 1, 40}] (* A213757 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213758 *)

T(n,k) = 5*T(n,k-1)-9*T(n,k-2)+7*T(n,k-3)-2*T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = x*(2*n - 1 - (2*n - 3)*x) and g(x) = (1 - 2*x)(1 - x )^3.

A213768 Rectangular array: (row n) = b**c, where b(h) = F(h), c(h) = 2n-3+2h, F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 4, 3, 10, 8, 5, 21, 18, 12, 7, 40, 35, 26, 16, 9, 72, 64, 49, 34, 20, 11, 125, 112, 88, 63, 42, 24, 13, 212, 191, 152, 112, 77, 50, 28, 15, 354, 320, 257, 192, 136, 91, 58, 32, 17, 585, 530, 428, 323, 232, 160, 105, 66, 36, 19

Offset: 1

Clark Kimberling, Jun 21 2012

Principal diagonal:  A213769.
Antidiagonal sums:  A213770.
Row 1,  (1,1,2,3,5,...)**(1,3,5,7,9,...): A001891.
Row 2,  (1,1,2,3,5,...)**(3,5,7,9,11,...).
Row 3,  (1,1,2,3,5,...)**(5,7,9,11,13,...).
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....4....10...21...40....72....125
3....8....18...35...64....112...191
5....12...26...49...88....152...257
7....16...34...63...112...192...323
9....20...42...77...136...232...389
11...24...50...91...160...272...455

Clark Kimberling, Antidiagonals n=1..80, flattened

Cf. A001891, A213500, A213769, A213770.

b[n_] := Fibonacci[n]; c[n_] := 2 n - 1;
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}]  (* A213768 *)
Table[t[n, n], {n, 1, 40}] (* A213769 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213770 *)

T(n,k) = 3*T(n,k-1)-2*T(n,k-2)-T(n,k-3)+T(n,k-4).
G.f. for row n:  f(x)/g(x), where f(x) = x*(2*n - 1 - (2*n - 3)*x) and g(x) = (1 - x - x^2)(1 - x )^2.
T(n,k) = 2*n*Fibonacci(k+2) + Lucas(k+2) - 2*(k+n) - 3. - Ehren Metcalfe, Jul 08 2019

cp's OEIS Frontend

A213564 Rectangular array: (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.

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A213569 Principal diagonal of the convolution array A213568.

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A213571 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.

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A213584 Rectangular array: (row n) = b**c, where b(h) = F(h+1), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

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A213586 Antidiagonal sums of the convolution array A213584.

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A213750 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 2*(n-1+h)-1, n>=1, h>=1, and ** = convolution.

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A213571 Rectangular array: (row n) = bc, where b(h) = h, c(h) = (n-1+h)^2, n>=1, h>=1, and = convolution.

A213584 Rectangular array: (row n) = bc, where b(h) = F(h+1), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

A213756 Rectangular array: (row n) = b**c, where b(h) = -1 + 2^h, c(h) = 2n - 3 + 2h, n>=1, h>=1, and ** = convolution.

A213768 Rectangular array: (row n) = b**c, where b(h) = F(h), c(h) = 2n-3+2h, F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.