A213500 -id:A213500 - cp's OEIS frontend

A213547 Antidiagonal sums of the convolution array A213505.

1, 12, 68, 260, 777, 1960, 4368, 8856, 16665, 29524, 49764, 80444, 125489, 189840, 279616, 402288, 566865, 784092, 1066660, 1429428, 1889657, 2467256, 3185040, 4069000, 5148585, 6456996, 8031492, 9913708, 12149985, 14791712, 17895680, 21524448, 25746721, 30637740

Offset: 1

Clark Kimberling, Jun 16 2012

Also, the antidiagonal sums of the convolution array A213555.
An m-star is an m-antichain with a smallest element adjoined. Then, a(n) is the number of proper mergings of a 2-star and an (n-1)-chain, see example. - Henri Mühle, Jan 23 2013
Convolution of A000290 and A000578. - Stefano Spezia, Apr 07 2023

			From _Henri Mühle_, Jan 23 2013: (Start)
For n=2, let S=({s0,s1,s2},{(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2)}) be a 2-star, and let C=({c},{(c,c)}) be a 1-chain. The a(2)=12 proper mergings of S and C are:
({s0,s1,s2,c},{(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(c,s0),(c,s1),(c,s2),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(c,s1),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(c,s2),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(c,s1),(c,s2),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(s0,c),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(s0,c),(c,s1),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(s0,c),(c,s2),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(s0,c),(c,s1),(c,s2),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(s1,c),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(s2,c),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
({s0,s1,s2,c},{(s1,c),(s2,c),(s0,s0),(s0,s1),(s0,s2),(s1,s1),(s2,s2),(c,c)})
(End)

Clark Kimberling, Table of n, a(n) for n = 1..1000
Henri Muehle, Proper Mergings of Stars and Chains are Counted by Sums of Antidiagonals in Certain Convolution Arrays -- The Details, arXiv preprint arXiv:1301.1654 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000290, A000578, A213500, A213505.

Mathematica
```
(See A213505.)
```

{a(n) = n++; (n^6 - n^2) / 60}; /* Michael Somos, Oct 08 2017 */

a(n) = (n^6 + 6*n^5 + 15*n^4 + 20*n^3 + 14*n^2 + 4*n)/60.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: x*(1+x)*(1+4*x+x^2)/(1-x)^7.
a(n) = a(-2-n) and a(n-1) = (n^6 - n^2) / 60 for all n in Z. - Michael Somos, Oct 08 2017
E.g.f.: exp(x)*x*(60 + 300*x + 350*x^2 + 140*x^3 + 21*x^4 + x^5)/60. - Stefano Spezia, Apr 07 2023

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

A213550 Rectangular array: (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 5, 2, 15, 9, 3, 35, 25, 13, 4, 70, 55, 35, 17, 5, 126, 105, 75, 45, 21, 6, 210, 182, 140, 95, 55, 25, 7, 330, 294, 238, 175, 115, 65, 29, 8, 495, 450, 378, 294, 210, 135, 75, 33, 9, 715, 660, 570, 462, 350, 245, 155, 85, 37, 10, 1001, 935, 825, 690, 546

Offset: 1

Clark Kimberling, Jun 16 2012

Principal diagonal:  A002418
Antidiagonal sums:  A005585
row 1,  (1,3,6,...)**(1,2,3,...):  A000332
row 2,  (1,3,6,...)**(2,3,4,...):  A005582
row 3,  (1,3,6,...)**(3,4,5,...):  A095661
row 4,  (1,3,6,...)**(4,5,6,...):  A095667
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....5....15...35....70....126
2....9....25...55....105...182
3....13...35...75....140...238
4....17...45...95....175...294
5....21...55...115...210...350

Cf. A213500, A213548.

b[n_] := n (n + 1)/2; 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}]  (* A213550 *)
d = Table[t[n, n], {n, 1, 40}] (* A002418 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A005585 *)

T(n,k) = 5*T(n,k-1) - 10*T(n,k-2) + 10*T(n,k-3) - 5*T(n,k-4) + T(n,k-5).

G.f. for row n: f(x)/g(x), where f(x) = n-(n-1)*x and g(x) = (1 - x)^5.

A213579 Rectangular array: (row n) = bc, where b(h) = F(h), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 3, 2, 7, 5, 3, 14, 11, 7, 4, 26, 21, 15, 9, 5, 46, 38, 28, 19, 11, 6, 79, 66, 50, 35, 23, 13, 7, 133, 112, 86, 62, 42, 27, 15, 8, 221, 187, 145, 106, 74, 49, 31, 17, 9, 364, 309, 241, 178, 126, 86, 56, 35, 19, 10, 596, 507, 397, 295, 211, 146, 98, 63, 39, 21

Offset: 1

Clark Kimberling, Jun 18 2012

Principal diagonal: A213580.
Antidiagonal sums: A053808.
Row 1, (1,1,2,3,5,...)**(1,2,3,4,...): A001924.
Row 2, (1,1,2,3,5,...)**(2,3,4,5,...): A023548.
Row 3, (1,1,2,3,5,...)**(3,4,5,6,...): A023552.
Row 4, (1,1,2,3,5,...)**(4,5,6,7,...): A210730.
Row 5, (1,1,2,3,5,...)**(5,6,7,8,...): A210731.
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....3....7....14...26...46
2....5....11...21...38...66
3....7....15...28...50...86
4....9....19...35...62...106
5....11...23...42...74...126
6....13...27...49...86...146

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

Cf. A000045, A213500.

Flat(List([1..12], n-> List([1..n], k-> Fibonacci(k+3) + n*Fibonacci(k+2) -(n+k+2) ))); # G. C. Greubel, Jul 08 2019

[[Fibonacci(k+3) + n*Fibonacci(k+2) -(n+k+2): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 08 2019

(* First program *)
b[n_]:= Fibonacci[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}]] (* A213579 *)
r[n_]:= Table[T[n, k], {k, 40}]
d = Table[T[n, n], {n, 1, 40}] (* A213580 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A053808 *)
(* Second program *)
Table[Fibonacci[n-k+4] +k*Fibonacci[n-k+3] -(n+3), {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 08 2019 *)

t(n,k) = fibonacci(n-k+4) + k*fibonacci(n-k+3) - (n+3);
for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 08 2019

[[fibonacci(k+3) + n*fibonacci(k+2) -(n+k+2) 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 - (n-1)*x and g(x) = (1-x-x^2) *(1-x)^2.
T(n, k) = Fibonacci(k+3) + n*Fibonacci(k+2) - (n+k+2). - G. C. Greubel, Jul 08 2019

A213587 Rectangular array: (row n) = bc, where b(h) = F(h+1), c(h) = F(n+h), F = A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 4, 2, 10, 7, 3, 22, 17, 11, 5, 45, 37, 27, 18, 8, 88, 75, 59, 44, 29, 13, 167, 146, 120, 96, 71, 47, 21, 310, 276, 234, 195, 155, 115, 76, 34, 566, 511, 443, 380, 315, 251, 186, 123, 55, 1020, 931, 821, 719, 614, 510, 406, 301, 199, 89, 1819, 1675, 1497, 1332, 1162, 994, 825, 657, 487, 322, 144

Offset: 1

Clark Kimberling, Jun 19 2012

Principal diagonal: A213588.
Antidiagonal sums: A213589.
Row 1,  (1,2,3,5,...)**(1,2,3,5,...): A004798.
Row 2,  (1,2,3,5,...)**(2,3,5,8,...)
Row 3,  (1,2,3,5,...)**(3,5,8,13,...)
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
  1....4....10....22....45....88....167
  2....7....17....37....75....146...276
  3....11...27....59....120...234...443
  5....18...44....96....195...380...719
  8....29...71....155...315...614...1162
  13...47...115...251...510...994...1881

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

Cf. A213500, A213588, A213589, A004798.

Flat( List([1..12], n-> List([1..n], k-> ((n-k+1)*Lucas(1,-1, n+3)[2] - Fibonacci(n-k+1)*Lucas(1,-1,k-1)[2])/5 ))); # G. C. Greubel, Jul 08 2019

[[((n-k+1)*Lucas(n+3) - Fibonacci(n-k+1)*Lucas(k-1))/5: k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 08 2019

(* First program *)
b[n_]:= Fibonacci[n+1]; c[n_]:= Fibonacci[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}]] (* A213587 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213588 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213589 *)
(* Second program *)
Table[((n-k+1)*LucasL[n+3] - Fibonacci[n-k+1]*LucasL[k-1])/5, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 08 2019 *)

lucas(n) = fibonacci(n+1) + fibonacci(n-1);
t(n,k) = ((n-k+1)*lucas(n+3) - fibonacci(n-k+1)*lucas(k-1))/5;
for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 08 2019

[[((n-k+1)*lucas_number2(n+3,1,-1) - fibonacci(n-k+1)* lucas_number2(k-1, 1,-1))/5 for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 08 2019

Rows: T(n,k) = 2*T(n,k-1) + T(n,k-2) - 2*T(n,k-3) - T(n,k-4).
Columns: T(n,k) = T(n-1,k) + T(n-2,k).
G.f. for row n: f(x)/g(x), where f(x) = F(n+1) + F(n+2)*x + F(n)*x^2 and g(x) = (1 - x - x^2)^2.
T(n, k) = (k*Lucas(n+k+2) - Fibonacci(k)*Lucas(n-1))/5. - G. C. Greubel, Jul 08 2019

A213590 Rectangular array: (row n) = bc, where b(h) = h^2, c(h) = F(n-1+h), F = A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 5, 1, 15, 6, 2, 36, 20, 11, 3, 76, 51, 35, 17, 5, 148, 112, 87, 55, 28, 8, 273, 224, 188, 138, 90, 45, 13, 485, 421, 372, 300, 225, 145, 73, 21, 839, 758, 694, 596, 488, 363, 235, 118, 34, 1424, 1324, 1243, 1115, 968, 788, 588, 380, 191, 55, 2384, 2263, 2163, 2001, 1809, 1564, 1276, 951, 615, 309, 89

Offset: 1

Clark Kimberling, Jun 19 2012

Principal diagonal: A213504.
Antidiagonal sums: A213557.
Row 1,  (1,4,9,16,...)**(1,1,2,3,5,...): A053808.
Row 2,  (1,4,9,16,...)**(1,2,3,5,8,...): A213586.
Row 3,  (1,4,9,16,...)**(2,3,5,8,13,...).
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....5....15....36....76.....148
1....6....20....51....112....224
2....11...35....87....188....372
3....17...55....138...300....596
5....28...90....225...488....868
8....45...145...363...788....1564
13...73...235...588...1276...2532

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

Cf. A213500, A213587.

F:=Fibonacci;; Flat(List([1..12],n-> List([1..n],k-> F(n+7)-F(k+6) -2*(n-k+1)*F(k+3)-(n-k+1)^2*F(k+1) ))) # G. C. Greubel, Jul 05 2019

F:=Fibonacci; [[F(n+7) -F(k+6) -2*(n-k+1)*F(k+3) -(n-k+1)^2 *F(k+1): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 05 2019

(* First program *)
b[n_]:= n^2; c[n_]:= Fibonacci[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}]] (* A213590 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213504 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213557 *)
(* Second program *)
t[n_, k_]:= Fibonacci[n+7] - Fibonacci[k+6] - 2*(n-k+1)*Fibonacci[k+3] - (n-k+1)^2*Fibonacci[k+1]; Table[t[n, k], {n, 1, 12}, {k, 1, n}]//Flatten (* G. C. Greubel, Jul 05 2019 *)

f=fibonacci; t(n,k) = f(n+7) -f(k+6) -2*(n-k+1)*f(k+3) -(n-k+1)^2 *f(k+1);
for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 05 2019

f=fibonacci; [[f(n+7) -f(k+6) -2*(n-k+1)*f(k+3) - (n-k+1)^2* f(k+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 05 2019

Rows:  T(n,k) = 4*T(n,k-1) -5*T(n,k-2) +*T(n,k-3) +2*T(n,k-4) -T(n,k-5).
Columns:  T(n,k) = T(n-1,k) + T(n-2,k).
G.f. for row n:  f(x)/g(x), where f(x) = F(n) + F(n+1)*x + F(n-1)*x^2 and g(x) = (1 - x - x^2)*(1 - x )^3.
T(n, k) = Fibonacci(n+k+6) - Fibonacci(n+6) - 2*k*Fibonacci(n+3) - k^2*Fibonacci(n+1). - G. C. Greubel, Jul 05 2019

A213548 Rectangular array: (row n) = bc, where b(h) = h, c(h) = m(m+1)/2, m = n-1+h, n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 5, 3, 15, 12, 6, 35, 31, 22, 10, 70, 65, 53, 35, 15, 126, 120, 105, 81, 51, 21, 210, 203, 185, 155, 115, 70, 28, 330, 322, 301, 265, 215, 155, 92, 36, 495, 486, 462, 420, 360, 285, 201, 117, 45, 715, 705, 678, 630, 560, 470, 365, 253, 145, 55, 1001

Offset: 1

Clark Kimberling, Jun 16 2012

Principal diagonal: A213549.
Antidiagonal sums: A051836.
Row 1, (1,2,3,...)**(1,3,6,...): A000332.
Row 2, (1,2,3,...)**(3,6,10,...): A005718.
Row 3, (1,2,3,...)**(6,10,15,...): k*(k+1)*(k^2 + 13*k + 58)/24.
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
.  1,  5,  15,  35,  70, ...
.  3, 12,  31,  65, 120, ...
.  6, 22,  53, 105, 185, ...
. 10, 35,  81, 155, 265, ...
. 15, 51, 115, 215, 360, ...
. 21, 70, 155, 285, 470, ...
...
T(5,1) = (1)**(15) = 15;
T(5,2) = (1,2)**(15,21) = 1*21 + 2*15 = 51;
T(5,3) = (1,2,3)**(15,21,28) = 1*28 + 2*21 + 3*15 = 115;
T(4,4) = (1,2,3,4)**(10,15,21,28) = 1*28 + 2*21 + 3*15 + 4*10 = 155.

Clark Kimberling, Antidiagonals n = 1..60

Cf. A213500.

b[n_] := n; c[n_] := n (n + 1)/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}]  (* A213548 *)
d = Table[t[n, n], {n, 1, 40}] (* A213549 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A051836 *)

T(n,k) = 5*T(n,k-1) - 10*T(n,k-2) + 10*T(n,k-3) - 5*T(n,k-4) + T(n,k-5).

G.f. for row n: f(x)/g(x), where f(x) = n*(n+1) - 2*((n-1)^2)*x + n*(n-1)*x^2 and g(x) = 2*(1 - x)^5.

A213553 Rectangular array: (row n) = bc, where b(h) = h, c(h) = (n-1+h)^3, n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 10, 8, 46, 43, 27, 146, 142, 118, 64, 371, 366, 334, 253, 125, 812, 806, 766, 658, 466, 216, 1596, 1589, 1541, 1406, 1150, 775, 343, 2892, 2884, 2828, 2666, 2346, 1846, 1198, 512, 4917, 4908, 4844, 4655, 4271, 3646, 2782, 1753, 729, 7942

Offset: 1

Clark Kimberling, Jun 17 2012

Principal diagonal: A213554
Antidiagonal sums: A101089
row 1, (1,2,3,...)**(1,8,27,...): A024166
row 2, (1,2,3,...)**(8,27,64,...): (3*k^5 + 30*k^4 + 115*k^3 + 210*k^2 + 122*k)/60
row 3, (1,2,3,...)**(27,64,125,...): (3*k^5 + 45*k^4 + 265*k^3 + 765*k^2 + 542*k)/120
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1.....10....46.....146....371
8.....43....142....366....806
27....118...334....766....1541
64....253...658....1406...2666
125...466...1150...2346...4271

G. C. Greubel, Antidiagonals n = 1..100, flattened

Cf. A213500.

Flat(List([1..12], n-> List([1..n], k-> Binomial(n-k+2, 2)*(12*k^3 +9*k^2*n -9*k^2 +6*k*n^2 +3*k*n -k +n*(3*n^2 +6*n +1))/30 ))); # G. C. Greubel, Jul 31 2019

[Binomial(n-k+2, 2)*(12*k^3 +9*k^2*n -9*k^2 +6*k*n^2 +3*k*n -k + n*(3*n^2 +6*n +1))/30: k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 31 2019

(* First program *)
b[n_]:= n; c[n_]:= n^3;
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}]  (* A213553 *)
d = Table[T[n, n], {n, 1, 40}] (* A213554 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A101089 *)
(* Second program *)
Table[Binomial[n-k+2, 2]*(12*k^3 +9*k^2*n -9*k^2 +6*k*n^2 +3*k*n -k +n*(3*n^2 +6*n +1))/30, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 31 2019 *)

t(n,k) = binomial(n-k+2, 2)*(12*k^3 +9*k^2*n -9*k^2 +6*k*n^2 +3*k*n -k +n*(3*n^2 +6*n +1))/30;
for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 31 2019

[[binomial(n-k+2, 2)*(12*k^3 +9*k^2*n -9*k^2 +6*k*n^2 +3*k*n -k + n*(3*n^2 +6*n +1))/30 for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 31 2019

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^3 + (-3*n^3 + 3*n^2 + 3*n + 1)*x + (3*n^3 - 6*n^2 + 4)*x^2 - ((n-1)^3)*x^3 and g(x) = (1 - x)^6.
T(n,k) = k*((3*k^4 - 5*k^2 + 2) + 15*k*(k^2 - 1)*n + 30*(k^2 - 1)*n^2 + 30*(k + 1)*n^3)/60. - G. C. Greubel, Jul 31 2019

A213575 Antidiagonal sums of the convolution array A213573.

Original entry on oeis.org

1, 10, 47, 158, 441, 1098, 2539, 5590, 11909, 24818, 50967, 103662, 209521, 421786, 846947, 1697990, 3400893, 6807618, 13622095, 27252190, 54513641, 109037930, 218088027, 436189878, 872395381, 1744808338, 3489636359

Offset: 1

Clark Kimberling, Jun 18 2012

Clark Kimberling, Table of n, a(n) for n = 1..500
S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-9,2).

Cf. A213564, A213500, A232603, A232604.

List([1..30], n-> 13*2^(n+1)-(n^3+6*n^2+18*n+26)); # G. C. Greubel, Jul 25 2019

[13*2^(n+1)-(n^3+6*n^2+18*n+26): n in [1..30]]; // G. C. Greubel, Jul 25 2019

(* First program *)
b[n_]:= 2^(n-1); 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}]  (* A213573 *)
d = Table[t[n, n], {n, 1, 40}] (* A213574 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213575 *)
(* Additional programs *)
Table[Sum[k^3*2^(n-k),{k,0,n}],{n,1,30}] (* Vaclav Kotesovec, Nov 28 2013 *)

vector(30, n, 13*2^(n+1)-(n^3+6*n^2+18*n+26)) \\ G. C. Greubel, Jul 25 2019

[13*2^(n+1)-(n^3+6*n^2+18*n+26) for n in (1..30)] # G. C. Greubel, Jul 25 2019

a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 9*a(n-4) + 2*a(n-5).
G.f.: x*(1 + 4 x + x^2)/((1 - 2*x)*(1 - x)^4).
From Stanislav Sykora, Nov 27 2013: (Start)
a(n) = 2^n*Sum_{k=0..n} k^p*q^k, for p=3, q=1/2.
a(n) = 2^(n+1)*13 - (n^3 + 6*n^2 + 18*n + 26). (End)
a(n) = 2*a(n-1) + n^3. - Sochima Everton, Biereagu, Jul 14 2019
E.g.f.: 26*exp(2*x) - (26 +25*x +9*x^2 +x^3)*exp(x). - G. C. Greubel, Jul 25 2019

A213778 Rectangular array: (row n) = bc, where b(h) = h, c(h) = 1+[(n-1+h)/2], n>=1, h>=1, [ ] = floor, and = convolution.

Original entry on oeis.org

1, 4, 2, 9, 6, 2, 17, 13, 7, 3, 28, 23, 15, 9, 3, 43, 37, 27, 19, 10, 4, 62, 55, 43, 33, 21, 12, 4, 86, 78, 64, 52, 37, 25, 13, 5, 115, 106, 90, 76, 58, 43, 27, 15, 5, 150, 140, 122, 106, 85, 67, 47, 31, 16, 6, 191, 180, 160, 142, 118, 97, 73, 53, 33, 18, 6, 239

Offset: 1

Clark Kimberling, Jun 21 2012

Principal diagonal: A213779.
Antidiagonal sums: A213780.
Row 1,  (1,2,3,4,5,...)**(1,2,2,3,3,4,4,...): A005744.
Row 2,  (1,2,3,4,5,...)**(2,2,3,3,4,4,...)
Row 3,  (1,2,3,4,5,...)**(3,4,4,5,5,...)
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1...4....9....17...28...43....62
2...6....13...23...37...55....78
2...7....15...27...43...64....90
3...9....19...33...52...76....106
3...10...21...37...58...85....118
4...12...25...43...67...97....134
4...13...27...47...73...106...146

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

Cf. A213500.

b[n_] := n; c[n_] := 1 + Floor[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}]  (* A213778 *)
Table[t[n, n], {n, 1, 40}] (* A213779 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213780 *)

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

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

cp's OEIS Frontend

A213547 Antidiagonal sums of the convolution array A213505.

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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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A213550 Rectangular array: (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

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A213579 Rectangular array: (row n) = b**c, where b(h) = F(h), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

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A213587 Rectangular array: (row n) = b**c, where b(h) = F(h+1), c(h) = F(n+h), F = A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

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A213590 Rectangular array: (row n) = b**c, where b(h) = h^2, c(h) = F(n-1+h), F = A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

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

A213579 Rectangular array: (row n) = bc, where b(h) = F(h), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

A213587 Rectangular array: (row n) = bc, where b(h) = F(h+1), c(h) = F(n+h), F = A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

A213590 Rectangular array: (row n) = bc, where b(h) = h^2, c(h) = F(n-1+h), F = A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

A213548 Rectangular array: (row n) = bc, where b(h) = h, c(h) = m(m+1)/2, m = n-1+h, n>=1, h>=1, and = convolution.

A213553 Rectangular array: (row n) = bc, where b(h) = h, c(h) = (n-1+h)^3, n>=1, h>=1, and = convolution.

A213778 Rectangular array: (row n) = bc, where b(h) = h, c(h) = 1+[(n-1+h)/2], n>=1, h>=1, [ ] = floor, and = convolution.