Shara Lalo - cp's OEIS frontend

A319095 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1, k) + T(n-1, k-1) + T(n-1, k-2) + 3T(n-1, k-3) + T(n-1, k-4) + T(n-1, k-5) + T(n-1, k-6) + T(n-1, k-7) for k = 0..7n; T(n,k) = 0 for n or k < 0.

1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 3, 8, 9, 10, 15, 12, 11, 10, 9, 4, 3, 2, 1, 1, 3, 6, 16, 27, 39, 64, 78, 90, 108, 108, 102, 94, 84, 60, 46, 33, 21, 10, 6, 3, 1, 1, 4, 10, 28, 59, 104, 188, 288, 401, 556, 686, 796, 899, 944, 928, 880, 803, 668, 542, 420, 305, 200, 132, 80, 43, 20, 10, 4, 1

Offset: 0

Shara Lalo, Oct 01 2018

Row n gives the coefficients in the expansion of (1 + x + x^2 + 3*x^3 + x^4 + x^5 + x^6 + x^7)^n, where n is a nonnegative integer. The row sum at row n is s(n) = 10^n. In the center-justified triangle, the sum of numbers along "first layer" skew diagonals pointing top-right are the coefficients in the expansion of 1/(1 - x - x^2 - x^3 - 3*x^4 - x^5 - x^6 - x^7 - x^8) and the sum of numbers along "first layer" skew diagonals pointing top-left are the coefficients in the expansion of 1/(1 - x - x^2 - x^3 - x^4 - 3*x^5 - x^6 - x^7 - x^8), see links.

Note: Coefficients in expansion of (1 + x + ... + x^7)^n is given in A171890 (Octonomial coefficient array).

			Triangle begins:
1;
1, 1, 1,  3,  1,  1,  1,  1;
1, 2, 3,  8,  9, 10, 15, 12, 11,  10,   9,   4,  3,  2,  1;
1, 3, 6, 16, 27, 39, 64, 78, 90, 108, 108, 102, 94, 84, 60, 46, 33, 21, 10, 6, 3, 1;
...

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

Cf. A171890.

f[n_] := CoefficientList[(1 + x + x^2 + 3*x^3 + x^4 + x^5 + x^6 + x^7)^n, x] ; Join @@ Table[f[k], {k, 0, 5}] // Flatten (* Amiram Eldar, Dec 07 2018 *)

T(n, k) := ratcoef((1 + x + x^2 + 3*x^3 + x^4 + x^5 + x^6 + x^7)^n, x, k)$
create_list(T(n,k), n, 0, 10, k, 0, 7*n); /* Franck Maminirina Ramaharo, Nov 27 2018 */

row(n) = Vecrev((1 + x + x^2 + 3*x^3 + x^4 + x^5 + x^6 + x^7)^n);
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Oct 15 2018

T(n,k) = Sum_{i=0..k} Sum_{j=2*i..k} Sum_{q=3*i..k} Sum_{r=4*i..k} Sum_{p=5*i..k} Sum_{d=6*i..k}(f) for k=0..7*n; f= (3^(q - 2*r + p)*n!)/((n + d - k)!*(k + p - 2*d)!*(d + r - 2*p)!*(q + p - 2*r)!*(j + r - 2*q)!*(i + q - 2*j)!*(j - 2*i)!*i!); f=0 for (n + d - k)<0 or (k + p - 2*d)<0 or (d + r - 2*p)<0 or (q + p - 2*r)<0 or (j + r - 2*q)<0 or (i + q - 2*j)<0 or (j - 2*i)<0. A novel formula proven by Shara Lalo and Zagros Lalo. Also see formula in Links section.

G.f.: 1/(1 - t*x - t*x^2 - t*x^3 - 3*t*x^4 - t*x^5 - t*x^6 - t*x^7 - t*x^8).

A319094 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1, k) + T(n-1, k-1) + 2 T(n-1, k-2) + T(n-1, k-3) + T(n-1, k-4) + T(n-1, k-5) + T(n-1, k-6) for k = 0..6*n; T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 5, 6, 8, 8, 9, 8, 7, 4, 3, 2, 1, 1, 3, 9, 16, 27, 36, 47, 54, 60, 58, 54, 45, 37, 27, 18, 10, 6, 3, 1, 1, 4, 14, 32, 65, 108, 166, 228, 296, 352, 396, 412, 409, 380, 336, 276, 215, 156, 108, 68, 39, 20, 10, 4, 1, 1, 5, 20, 55, 130, 256, 455, 725, 1075, 1475, 1907, 2310, 2655, 2885

Offset: 0

Shara Lalo, Oct 01 2018

Row n gives the coefficients in the expansion of (1 + x + 2*x^2 + x^3 + x^4 + x^5 + x^6)^n, where n is a nonnegative integer. The row sum at row n is s(n) = 8^n. In the center-justified triangle, the sum of numbers along "first layer" skew diagonals pointing top-right are the coefficients in the expansion of 1/(1 - x - x^2 - 2*x^3 - x^4 - x^5 - x^6 - x^7) and the sum of numbers along "first layer" skew diagonals pointing top-left are the coefficients in the expansion of 1/(1 - x - x^2 - x^3 - x^4 - 2*x^5 - x^6 - x^7), see links.

			Triangle begins:
1;
1, 1, 2,  1,  1,  1,  1;
1, 2, 5,  6,  8,  8,  9,  8,  7,  4,  3,  2,  1;
1, 3, 9, 16, 27, 36, 47, 54, 60, 58, 54, 45, 37, 27, 18, 10, 6, 3, 1;
...

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

Cf. A319092, A319093, A319095.

Clear[t, n, k];t[n_, k_] := t[n, k] = Sum[(2^(q - 2*r + p)*n!)/((n + p - k)!*(k + r - 2*p)!*(q - 2*r + p)!*(j - 2*q +r)!*(i - 2*j + q)!*(j - 2*i)!*i!), {i, 0, k}, {j, 2*i,k}, {q, 3*i, k}, {r, 4*i, k}, {p, 5*i, k}]; Flatten[Table[t[n, k], {n, 0, 3}, {k, 0, 6*n}]]
t[n_, k_] := t[n, k] = Sum[(2^(q - 2*r + p)*n!)/((n + p - k)!*(k + r - 2*p)!*(q - 2*r + p)!*(j - 2*q + r)!*(i - 2*j + q)!*(j - 2*i)!*i!), {i, 0, k}, {j, 0, k}, {q, 0, k}, {r, 0, k}, {p, 0, k}]; Table[t[n, k], {n, 0, 3}, {k, 0, 6*n}] // Flatten

row(n) = Vecrev((1 + x + 2*x^2 + x^3 + x^4 + x^5 + x^6)^n);
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Oct 15 2018

T(n,k) = Sum_{i=0..k} Sum_{j=2*i..k} Sum_{q=3*i..k} Sum_{r=4*i..k} Sum_{p=5*i..k }(f) for k=0..6*n; f = (2^(q - 2*r + p)*n!)/((n + p - k)!*(k + r - 2*p)!*(q - 2*r + p)!*(j - 2*q + r)!*(i - 2*j + q)!*(j - 2*i)!*i!); f=0 for (n + p - k)<0 or (k + r - 2*p)<0 or (q - 2*r + p)<0 or (j - 2*q + r)<0 or (i - 2*j + q)<0 or (j - 2*i)<0.

G.f.: 1/(1 - t*x - t*x^2 - 2*t*x^3 - t*x^4 - t*x^5 - t*x^6 - t*x^7).

A319093 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1, k) + T(n-1, k-1) - T(n-1, k-2) + 2T(n-1, k-3) + T(n-1, k-4) for k = 0..4n; T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 1, 1, -1, 2, 1, 1, 2, -1, 2, 7, -2, 2, 4, 1, 1, 3, 0, 1, 15, 3, -4, 24, 6, -1, 9, 6, 1, 1, 4, 2, 0, 23, 20, -14, 48, 55, -24, 46, 52, 2, 12, 20, 8, 1, 1, 5, 5, 0, 30, 51, -15, 60, 180, -25, 49, 280, 15, 30, 180, 72, 15, 45, 35, 10, 1, 1, 6, 9, 2, 36, 96, 11, 54, 387, 116, -51, 774, 376, -162, 804, 532

Offset: 0

Shara Lalo, Oct 01 2018

Row n gives the coefficients in the expansion of (1 + x - x^2 + 2*x^3 + x^4)^n, where n is a nonnegative integer. The row sum at row n is s(n) = 4^n. In the center-justified triangle, the sum of numbers along "first layer" skew diagonals pointing top-right are the coefficients in the expansion of 1/(1 - x - x^2 + x^3 - 2*x^4 - x^5) and the sum of numbers along "first layer" skew diagonals pointing top-left are the coefficients in the expansion of 1/(1 - x - 2*x^2 + x^3 - x^4 - x^5), see links. The central coefficients are given in A319096.

			Triangle begins:
1;
1, 1, -1, 2,  1;
1, 2, -1, 2,  7, -2,  2,   4,   1;
1, 3,  0, 1, 15,  3, -4,  24,   6,  -1,  9,   6,  1;
1, 4,  2, 0, 23, 20, -14, 48,  55, -24, 46,  52,  2, 12,  20,  8,  1;
1, 5,  5, 0, 30, 51, -15, 60, 180, -25, 49, 280, 15, 30, 180, 72, 15, 45, 35, 10, 1;
...

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

Cf. A319096.

Clear[t, n, k]; t[n_, k_] := t[n, k] = Sum[((-1)^(i + q - 2*j)*2^(j - 2*i)*n!)/((n - k + q)!*(k + j - 2*q)!*(i + q - 2*j)!*(j - 2*i)!*i!), {i, 0, k}, {j, 2*i, k}, {q, 3*i, k}]; Flatten[Table[t[n, k], {n, 0, 7}, {k, 0, 4*n}]]
Clear[t, n, k]; t[0, 0] = 1; t[n_, k_] :=  t[n, k] =   If[n < 0 || k < 0, 0,    t[n - 1, k] + t[n - 1, k - 1] - t[n - 1, k - 2] + 2 t[n - 1, k - 3] + t[n - 1, k - 4]]; Table[t[n, k], {n, 0, 6}, {k, 0, 4*n}  ]  // Flatten

row(n) = Vecrev((1 + x - x^2 + 2*x^3 + x^4)^n);
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Oct 15 2018

T(n,k) = Sum_{i=0..k} Sum_{j=2*i..k} Sum_{q=3*i..k}(f) for k = 0..4*n; f= (-1)^(i + q - 2*j)*2^(j - 2*i)*n!)/((n - k + q)!*(k + j - 2*q)!*(i + q - 2*j)!*(j - 2*i)!*i!); f=0 for (n - k + q)<0 or (k + j - 2*q)<0 or (i + q - 2*j) <0 or (j - 2*i) <0. A novel formula proven by Shara Lalo and Zagros Lalo. Also see formula in Links section.

G.f.: 1/(1 - t*x - t*x^2 + t*x^3 - 2*t*x^4 - t*x^5).

A319092 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1, k) + 2T(n-1, k-1) + 3T(n-1, k-2) + 4T(n-1, k-3) + 5T(n-1, k-4) + 6T(n-1, k-5) for k = 0..5n; T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 1, 4, 10, 20, 35, 56, 70, 76, 73, 60, 36, 1, 6, 21, 56, 126, 252, 441, 684, 954, 1204, 1365, 1344, 1169, 882, 540, 216, 1, 8, 36, 120, 330, 792, 1688, 3232, 5619, 8944, 13088, 17568, 21642, 24456, 25236, 23528, 19489, 14232, 8856, 4320, 1296, 1, 10, 55, 220, 715, 2002, 4970

Offset: 0

Shara Lalo, Oct 01 2018

Row n gives the coefficients in the expansion of (1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5)^n, where n is a nonnegative integer.
The row sum is s(n)=21^n (see A009965).
In the center-justified triangle, the sum of numbers along "first layer" skew diagonals pointing top-right are the coefficients in the expansion of 1/(1 - x - 2*x^2 -3*x^3 - 4*x^4 - 5*x^5 - 6*x^6) and the sum of numbers along "first layer" skew diagonals pointing top-left are the coefficients in the expansion of 1/(1 - 6*x - 5*x^2 - 4*x^3 - 3*x^4 - 2*x^5 - x^6), see links.

			Triangle begins:
1;
1, 2,  3,  4,   5,   6;
1, 4, 10, 20,  35,  56,  70,  76,  73,  60,    36;
1, 6, 21, 56, 126, 252, 441, 684, 954, 1204, 1365, 1344, 1169, 882, 540, 216;
...

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

Cf. A009965, A319093, A319094, A319095.

t[n_, k_] := t[n, k] = Sum[(2^(k + q - 2*r)*3^(j + r - 2*q)*4^(i + q - 2*j)*5^(j - 2*i)*6^i*n!)/((n - k + r)!*(k + q - 2*r)!*(j + r - 2*q)!*(i + q -2*j)!*(j - 2*i)!*i!), {i, 0, k}, {j, 2*i, k}, {q, 3*i, k}, {r, 4*i, k}]; Flatten[Table[t[n, k], {n, 0, 5}, {k, 0, 5 n}]]
t[0, 0] = 1; t[n_, k_] :=  t[n, k] =   If[n < 0 || k < 0, 0, t[n - 1, k] + 2 t[n - 1, k - 1] + 3 t[n - 1, k - 2] + 4 t[n - 1, k - 3] + 5 t[n - 1, k - 4] + 6 t[n - 1, k - 5]]; Table[t[n, k], {n, 0, 5}, {k, 0, 5 n}  ]  // Flatten

row(n) = Vecrev((1+2*x+3*x^2+4*x^3+5*x^4+6*x^5)^n);
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Oct 15 2018

T(n,k) = Sum_{i=0..k} Sum_{j=2*i..k} Sum_{q=3*i..k} Sum_{r=4*i..k}(f) for k=0..5*n; f=((2^(k + q - 2*r)*3^(j + r - 2*q)*4^(i + q - 2*j)*5^(j - 2*i)*6^i*n!)/((n - k + r)!*(k + q - 2*r)!*(j + r - 2*q)!*(i + q - 2*j)!*(j - 2*i)!*i!) ); f=0 for (n - k + r)<0 or (k + q - 2*r)<0; (j + r - 2*q)<0 or (i + q - 2*j) <0 or (j - 2*i)<0. A novel formula proven by Shara Lalo and Zagros Lalo. Also see formula in Links section.

G.f.: 1/(1 - x*t- 2*x^2*t - 3*x^3*t - 4*x^4*t - 5*x^5*t - 6*x^6*t).

A318686 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) - 2*T(n-1,k-2) + T(n-1,k-3) for k = 0..3n; T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 1, 0, -2, 1, 1, 0, -4, 2, 4, -4, 1, 1, 0, -6, 3, 12, -12, -5, 12, -6, 1, 1, 0, -8, 4, 24, -24, -26, 48, -8, -28, 24, -8, 1, 1, 0, -10, 5, 40, -40, -70, 120, 20, -150, 88, 40, -75, 40, -10, 1, 1, 0, -12, 6, 60, -60, -145, 240, 120, -460, 168, 360, -401, 48, 180, -154, 60, -12, 1, 1, 0, -14, 7, 84, -84, -259, 420, 350, -1085, 168, 1400, -1197, -504, 1342, -651, -252, 476, -273, 84, -14, 1

Offset: 0

Shara Lalo, Sep 06 2018

Row n gives coefficients in expansion of (1 - 2*x^2 + x^3)^n. Row sum s(n) = 1 when n = 0 and s(n) = 0 when n > 0, see link. In the center-justified triangle, the sum of numbers along "first layer" skew diagonals pointing top-right are the coefficients in expansion of 1/(1 - x + 2 x^3 - x^4) and the sum of numbers along "first layer" skew diagonals pointing top-left are the coefficients in expansion of 1/(1 - x + 2*x^2 - x^4), see links.

			Triangle begins:
1;
1, 0, -2, 1;
1, 0, -4, 2, 4, -4, 1;
1, 0, -6, 3, 12, -12, -5, 12, -6, 1;
1, 0, -8, 4, 24, -24, -26, 48, -8, -28, 24, -8, 1;
1, 0, -10, 5, 40, -40, -70, 120, 20, -150, 88, 40, -75, 40, -10, 1;
1, 0, -12, 6, 60, -60, -145, 240, 120, -460, 168, 360, -401, 48, 180, -154, 60, -12, 1;
...

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

t[0, 0] = 1; t[n_, k_] :=  t[n, k] =  If[n < 0 || k < 0, 0, t[n - 1, k] - 2 t[n - 1, k - 2] + t[n - 1, k - 3]]; Table[t[n, k], {n, 0, 7}, {k, 0, 3 n}  ]  // Flatten

T(0,0) = 1; T(n,k) = T(n-1,k) - 2*T(n-1,k-2) + T(n-1,k-3) for k = 0..3n; T(n,k)=0 for n or k < 0.

G.f.: 1/(1 - t + 2*t x^2 - t*x^3).

A318685 Triangle read by rows: T(0,0) = 1; T(n,k) = 2 T(n-1,k) - 3 T(n-1,k-1) + T(n-1,k-2) for k = 0..2n; T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 2, -3, 1, 4, -12, 13, -6, 1, 8, -36, 66, -63, 33, -9, 1, 16, -96, 248, -360, 321, -180, 62, -12, 1, 32, -240, 800, -1560, 1970, -1683, 985, -390, 100, -15, 1, 64, -576, 2352, -5760, 9420, -10836, 8989, -5418, 2355, -720, 147, -18, 1, 128, -1344, 6496, -19152, 38472, -55692, 59906, -48639, 29953, -13923, 4809, -1197, 203, -21, 1

Offset: 0

Shara Lalo, Sep 06 2018

Row n gives coefficients in expansion of (2 - 3*x + x^2)^n. Row sum s(n)= 1 when n = 0 and s(n)= 0 when n > 0, see link. In the center-justified triangle, the sum of numbers along "first layer" skew diagonals pointing top-right are the coefficients in expansion of 1/(1 - 2*x + 3*x^2 - x^3) and the sum of numbers along "first layer" skew diagonals pointing top-left are the coefficients in expansion of 1/(1-x+3*x^2-2x^3), see links. The generating function of the central terms is 1/sqrt(1 + 6*x + x^2), signed version of Central Delannoy numbers A001850.

			Triangle begins:
1;
2, -3, 1;
4, -12, 13, -6, 1;
8, -36, 66, -63, 33, -9, 1;
16, -96, 248, -360, 321, -180, 62, -12, 1;
32, -240, 800, -1560, 1970, -1683, 985, -390, 100, -15, 1;
64, -576, 2352, -5760, 9420, -10836, 8989, -5418, 2355, -720, 147, -18, 1;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

Cf. A001850.

t[n_, k_] := t[n, k] = Sum[(2^(n - k + i)/(n - k + i)!)*((-3)^(k - 2*i)/(k - 2*i)!)*(1/i!)*n!, {i, 0, k}];
  Flatten[Table[t[n, k], {n, 0, 7}, {k, 0, 2*n}]]
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2*t[n - 1, k] - 3*t[n - 1, k - 1] + t[n - 1, k - 2]];
  Flatten[Table[t[n, k], {n, 0, 7}, {k, 0, 2*n}]]

T(0,0) = 1; T(n,k) = 2 T(n-1,k) - 3 T(n-1,k-1) + T(n-1,k-2) for k = 0..2n; T(n,k)=0 for n or k < 0.

G.f.: 1/(1 - 2*t + 3*t*x - t*x^2).

A317509 Coefficients in expansion of 1/(1 + x - 2*x^5).

Original entry on oeis.org

1, -1, 1, -1, 1, 1, -3, 5, -7, 9, -7, 1, 9, -23, 41, -55, 57, -39, -7, 89, -199, 313, -391, 377, -199, -199, 825, -1607, 2361, -2759, 2361, -711, -2503, 7225, -12743, 17465, -18887, 13881, 569, -26055, 60985, -98759, 126521

Offset: 0

Shara Lalo, Sep 04 2018

Coefficients in expansion of 1/(1 + x - 2*x^5) are given by the sum of numbers along "fourth Layer" skew diagonals pointing top-left in triangle A065109 ((2-x)^n) and by the sum of numbers along "fourth Layer" skew diagonals pointing top-right in triangle A303872 ((-1+2*x)^n), see links.

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

Shara Lalo, Fourth layer skew diagonals in center-justified triangle of coefficients in expansion of (2 - x)^n
Shara Lalo, Fourth layer skew diagonals in center-justified triangle of coefficients in expansion of (-1 + 2x)^n
Index entries for linear recurrences with constant coefficients, signature (-1,0,0,0,2).

Cf. A065109, A303872.

CoefficientList[Series[1/(1 + x - 2 x^5), {x, 0, 42}], x]
a[0] = 1; a[n_] := a[n] = If[n < 0, 0, - a[n - 1] + 2 * a[n - 5]]; Table[a[n], {n, 0, 42}] // Flatten
LinearRecurrence[{-1,0,0,0,2}, {1,-1,1,-1,1}, 43]

my(x='x+O('x^99)); Vec(1/(1+x-2*x^5)) \\ Altug Alkan, Sep 04 2018

a(0)=1, a(n) = -1 * a(n-1) + 2 * a(n-5) for n >= 0; a(n)=0 for n < 0.

A317506 Triangle read by rows: T(0,0) = 1; T(n,k) = 2 T(n-1,k) - T(n-4,k-1) for 0 <= k <= floor(n/4); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 2, 4, 8, 16, -1, 32, -4, 64, -12, 128, -32, 256, -80, 1, 512, -192, 6, 1024, -448, 24, 2048, -1024, 80, 4096, -2304, 240, -1, 8192, -5120, 672, -8, 16384, -11264, 1792, -40, 32768, -24576, 4608, -160, 65536, -53248, 11520, -560, 1, 131072, -114688, 28160, -1792, 10

Offset: 0

Shara Lalo, Aug 31 2018

The numbers in rows of the triangle are along "third layer" skew diagonals pointing top-right in center-justified triangle given in A065109 ((2-x)^n) and along "third layer" skew diagonals pointing top-left in center-justified triangle given in A303872 ((-1+2x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (2-x)^n and (-1+2x)^n are given in A133156 (coefficients of Chebyshev polynomials of the second kind) and A305098 respectively.) The coefficients in the expansion of 1/(1-2x+x^4) are given by the sequence generated by the row sums. The row sums give A008937. If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.83928675521416113... (A058265: Decimal expansion of the tribonacci constant t, the real root of x^3-x^2-x-1), when n approaches infinity.

			Triangle begins:
       1;
       2;
       4;
       8;
      16,      -1;
      32,      -4;
      64,     -12;
     128,     -32;
     256,     -80,     1;
     512,    -192,     6;
    1024,    -448,    24;
    2048,   -1024,    80;
    4096,   -2304,   240,    -1;
    8192,   -5120,   672,    -8;
   16384,  -11264,  1792,   -40;
   32768,  -24576,  4608,  -160;
   65536,  -53248, 11520,  -560,  1;
  131072, -114688, 28160, -1792, 10;
  262144, -245760, 67584, -5376, 60;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

Row sums give A008937.
Cf. A065109, A303872.
Cf. A133156, A305098.
Cf. A058265.

t[n_, k_] := t[n, k] = 2^(n - 4 k) * (-1)^k/((n - 4 k)! k!) * (n - 3 k)!; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/4]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 * t[n - 1, k] - t[n - 4, k - 1]]; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/4]}] // Flatten

T(n,k) = 2^(n - 4*k) * (-1)^k / ((n - 4*k)! k!) * (n - 3*k)! where n >= 0 and 0 <= k <= floor(n/4).

A317505 Triangle read by rows: T(0,0) = 1; T(n,k) = - T(n-1,k) - 2 T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, -1, 1, -1, 2, 1, -4, -1, 6, 1, -8, 4, -1, 10, -12, 1, -12, 24, -1, 14, -40, 8, 1, -16, 60, -32, -1, 18, -84, 80, 1, -20, 112, -160, 16, -1, 22, -144, 280, -80, 1, -24, 180, -448, 240, -1, 26, -220, 672, -560, 32, 1, -28, 264, -960, 1120, -192, -1, 30, -312, 1320, -2016, 672, 1, -32, 364, -1760, 3360, -1792, 64, -1, 34, -420, 2288, -5280, 4032, -448

Offset: 0

Shara Lalo, Aug 02 2018

The numbers in rows of the triangle are along "second layer" skew diagonals pointing top-left in center-justified triangle given in A065109 ((2-x)^n) and along "second layer" skew diagonals pointing top-right in center-justified triangle given in A303872 ((-1+2x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (2-x)^n and (-1+2x)^n are given in A133156 (coefficients of Chebyshev polynomials of the second kind) and A305098 respectively.) The coefficients in the expansion of 1/(1+x+2x^3) are given by the sequence generated by the row sums (see A077973).

			Triangle begins:
   1;
  -1;
   1;
  -1,   2;
   1,  -4;
  -1,   6;
   1,  -8,    4;
  -1,  10,  -12;
   1, -12,   24;
  -1,  14,  -40,     8;
   1, -16,   60,   -32;
  -1,  18,  -84,    80;
   1, -20,  112,  -160,    16;
  -1,  22, -144,   280,   -80;
   1, -24,  180,  -448,   240;
  -1,  26, -220,   672,  -560,    32;
   1, -28,  264,  -960,  1120,  -192;
  -1,  30, -312,  1320, -2016,   672;
   1, -32,  364, -1760,  3360, -1792,   64;
  -1,  34, -420,  2288, -5280,  4032, -448;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 139-141, 391-393.

Row sums give A077973.
Cf. A065109, A303872.
Cf. A133156, A305098.

t[n_, k_] := t[n, k] = (-1)^(n - 3k) * 2^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 19}, {k, 0, Floor[n/3]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, - t[n - 1, k] + 2 t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 19}, {k, 0, Floor[n/3]}] // Flatten

T(n,k) = (-1)^(n - 3k) * 2^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).

A317504 Triangle read by rows: T(0,0) = 1; T(n,k) = 2 T(n-1,k) - T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 2, 4, 8, -1, 16, -4, 32, -12, 64, -32, 1, 128, -80, 6, 256, -192, 24, 512, -448, 80, -1, 1024, -1024, 240, -8, 2048, -2304, 672, -40, 4096, -5120, 1792, -160, 1, 8192, -11264, 4608, -560, 10, 16384, -24576, 11520, -1792, 60, 32768, -53248, 28160, -5376, 280, -1, 65536, -114688, 67584, -15360, 1120, -12

Offset: 0

Shara Lalo, Aug 02 2018

The numbers in rows of the triangle are along "second layer" skew diagonals pointing top-right in center-justified triangle given in A065109 ((2-x)^n) and along "second layer" skew diagonals pointing top-left in center-justified triangle given in A303872 ((-1+2x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (2-x)^n and (-1+2x)^n are given in A133156 (coefficients of Chebyshev polynomials of the second kind) and A305098 respectively.) The coefficients in the expansion of 1/(1-2x+x^3) are given by the sequence generated by the row sums. The row sums give A000071 (Fibonacci numbers - 1). If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.61803398874989484... (A001622: Decimal expansion of Golden ratio (phi or tau) = (1 + sqrt(5))/2), when n approaches infinity.

			Triangle begins:
       1;
       2;
       4;
       8,      -1;
      16,      -4;
      32,     -12;
      64,     -32,      1;
     128,     -80,      6;
     256,    -192,     24;
     512,    -448,     80,      -1;
    1024,   -1024,    240,      -8;
    2048,   -2304,    672,     -40;
    4096,   -5120,   1792,    -160,     1;
    8192,  -11264,   4608,    -560,    10;
   16384,  -24576,  11520,   -1792,    60;
   32768,  -53248,  28160,   -5376,   280,   -1;
   65536, -114688,  67584,  -15360,  1120,  -12;
  131072, -245760, 159744,  -42240,  4032,  -84;
  262144, -524288, 372736, -112640, 13440, -448, 1;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 139-141, 391-393.

Row sums give A000071.
Cf. A065109, A303872.
Cf. A133156, A305098.
Cf. A001622.

t[n_, k_] := t[n, k] = 2^(n - 3k) * (-1)^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/3]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 * t[n - 1, k] - t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/3]}] // Flatten

T(n,k) = 2^(n - 3k) * (-2)^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).

cp's OEIS Frontend

User: Shara Lalo

A319095 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1, k) + T(n-1, k-1) + T(n-1, k-2) + 3*T(n-1, k-3) + T(n-1, k-4) + T(n-1, k-5) + T(n-1, k-6) + T(n-1, k-7) for k = 0..7*n; T(n,k) = 0 for n or k < 0.

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A319094 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1, k) + T(n-1, k-1) + 2 T(n-1, k-2) + T(n-1, k-3) + T(n-1, k-4) + T(n-1, k-5) + T(n-1, k-6) for k = 0..6*n; T(n,k)=0 for n or k < 0.

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A319093 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1, k) + T(n-1, k-1) - T(n-1, k-2) + 2*T(n-1, k-3) + T(n-1, k-4) for k = 0..4*n; T(n,k)=0 for n or k < 0.

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A319092 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1, k) + 2*T(n-1, k-1) + 3*T(n-1, k-2) + 4*T(n-1, k-3) + 5*T(n-1, k-4) + 6*T(n-1, k-5) for k = 0..5*n; T(n,k)=0 for n or k < 0.

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A318686 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) - 2*T(n-1,k-2) + T(n-1,k-3) for k = 0..3n; T(n,k)=0 for n or k < 0.

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A318685 Triangle read by rows: T(0,0) = 1; T(n,k) = 2 T(n-1,k) - 3 T(n-1,k-1) + T(n-1,k-2) for k = 0..2n; T(n,k)=0 for n or k < 0.

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A317509 Coefficients in expansion of 1/(1 + x - 2*x^5).

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A319095 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1, k) + T(n-1, k-1) + T(n-1, k-2) + 3T(n-1, k-3) + T(n-1, k-4) + T(n-1, k-5) + T(n-1, k-6) + T(n-1, k-7) for k = 0..7n; T(n,k) = 0 for n or k < 0.

A319093 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1, k) + T(n-1, k-1) - T(n-1, k-2) + 2T(n-1, k-3) + T(n-1, k-4) for k = 0..4n; T(n,k)=0 for n or k < 0.

A319092 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1, k) + 2T(n-1, k-1) + 3T(n-1, k-2) + 4T(n-1, k-3) + 5T(n-1, k-4) + 6T(n-1, k-5) for k = 0..5n; T(n,k)=0 for n or k < 0.