A013613 -id:A013613 - cp's OEIS frontend

A081580 Pascal-(1,5,1) array.

1, 1, 1, 1, 7, 1, 1, 13, 13, 1, 1, 19, 61, 19, 1, 1, 25, 145, 145, 25, 1, 1, 31, 265, 595, 265, 31, 1, 1, 37, 421, 1585, 1585, 421, 37, 1, 1, 43, 613, 3331, 6145, 3331, 613, 43, 1, 1, 49, 841, 6049, 17401, 17401, 6049, 841, 49, 1, 1, 55, 1105, 9955, 40105, 65527, 40105, 9955, 1105, 55, 1

Offset: 0

Paul Barry, Mar 23 2003

One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A016921, A081589, A081590. Coefficients of the row polynomials in the Newton basis are given by A013613.

			Square array begins as:
  1,  1,   1,    1,    1, ... A000012;
  1,  7,  13,   19,   25, ... A016921;
  1, 13,  61,  145,  265, ... A081589;
  1, 19, 145,  595, 1585, ... A081590;
  1, 25, 265, 1585, 6145, ...
The triangle begins as:
  1;
  1,  1;
  1,  7,    1;
  1, 13,   13,    1;
  1, 19,   61,   19,     1;
  1, 25,  145,  145,    25,     1;
  1, 31,  265,  595,   265,    31,     1;
  1, 37,  421, 1585,  1585,   421,    37,    1;
  1, 43,  613, 3331,  6145,  3331,   613,   43,    1;
  1, 49,  841, 6049, 17401, 17401,  6049,  841,   49,  1;
  1, 55, 1105, 9955, 40105, 65527, 40105, 9955, 1105, 55, 1; - _Philippe Deléham_, Mar 15 2014

Vincenzo Librandi, Rows n = 0..100, flattened
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Cf. Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081581 (m = 6), A081582 (m = 7), A143683 (m = 8).

Cf. A002532, A016921, A081589, A081590.

A081580:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A081580(n,k,5): k in [0..n], n in [0..12]]; // G. C. Greubel, May 26 2021

Table[Hypergeometric2F1[-k, k-n, 1, 6], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 6).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2021

Square array T(n, k) defined by T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 5*T(n-1, k-1) + T(n-1, k).
Rows are the expansions of (1+5*x)^k/(1-x)^(k+1).
From Paul Barry, Aug 28 2008: (Start)
Number triangle T(n,k) = Sum_{j=0..n-k} binomial(n-k,j)*binomial(k,j)*5^j.
Riordan array (1/(1-x), x*(1+5*x)/(1-x)). (End)
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 6). - Jean-François Alcover, May 24 2013
E.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(6*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 12*x + 36*x^2/2) = 1 + 13*x + 61*x^2/2! + 145*x^3/3! + 265*x^4/4! + 421*x^5/5! + .... - Peter Bala, Mar 05 2017
Sum_{k=0..n} T(n, k, 3) = A002532(n+1). - G. C. Greubel, May 26 2021

A038255 Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).

Original entry on oeis.org

1, 6, 1, 36, 12, 1, 216, 108, 18, 1, 1296, 864, 216, 24, 1, 7776, 6480, 2160, 360, 30, 1, 46656, 46656, 19440, 4320, 540, 36, 1, 279936, 326592, 163296, 45360, 7560, 756, 42, 1, 1679616, 2239488, 1306368, 435456, 90720, 12096, 1008

Offset: 0

N. J. A. Sloane

T(n,k) = A013613(n,n-k), 0 <= k <= n. - Reinhard Zumkeller, Nov 21 2013

			1
6, 1
36, 12, 1
216, 108, 18, 1
1296, 864, 216, 24, 1
7776, 6480, 2160, 360, 30, 1
46656, 46656, 19440, 4320, 540, 36, 1
279936, 326592, 163296, 45360, 7560, 756, 42, 1
1679616, 2239488, 1306368, 435456, 90720, 12096, 1008, 48, 1

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

Cf. A038207.

Cf. A000420 (row sums), A013613 (mirrored), A110440, A007318, A000400.

a038255 n k = a038255_tabl !! n !! k
a038255_row n = a038255_tabl !! n
a038255_tabl = map reverse a013613_tabl
-- Reinhard Zumkeller, Nov 21 2013

for i from 0 to 8 do seq(binomial(i, j)*6^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Dec 21 2007

Table[Binomial[n,m]6^(n-m),{n,0,10},{m,0,n}]//Flatten (* Harvey P. Dale, Dec 25 2019 *)

G.f.: 1/(1 - 6*x - x*y). - Ilya Gutkovskiy, Apr 21 2017

A013620 Triangle of coefficients in expansion of (2+3x)^n.

Original entry on oeis.org

1, 2, 3, 4, 12, 9, 8, 36, 54, 27, 16, 96, 216, 216, 81, 32, 240, 720, 1080, 810, 243, 64, 576, 2160, 4320, 4860, 2916, 729, 128, 1344, 6048, 15120, 22680, 20412, 10206, 2187, 256, 3072, 16128, 48384, 90720, 108864, 81648, 34992, 6561, 512

Offset: 0

N. J. A. Sloane

Row sums give A000351; central terms give A119309. - Reinhard Zumkeller, May 14 2006

			Triangle begins:
1;
2,3;
4,12,9;
8,36,54,27;
16,96,216,216,81;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Gábor Kallós, A generalization of Pascal’s triangle using powers of base numbers, Annales mathématiques Blaise Pascal, 13 no. 1 (2006), p. 1-15.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A038220, A000079, A000244, A013613.

a013620 n k = a013620_tabl !! n !! k
a013620_row n = a013620_tabl !! n
a013620_tabl = iterate (\row ->
   zipWith (+) (map (* 2) (row ++ [0])) (map (* 3) ([0] ++ row))) [1]
-- Reinhard Zumkeller, May 26 2013, Apr 02 2011

Flatten[Table[Binomial[i, j] 2^(i-j) 3^j, {i, 0, 10}, {j, 0, i}]] (* Vincenzo Librandi, Apr 22 2014 *)

G.f.: 1 / [1 - x(2+3y)].

T(n,k) = A007318(n,k) * A036561(n,k). - Reinhard Zumkeller, May 14 2006

A038220 Triangle whose (i,j)-th entry is binomial(i,j)3^(i-j)2^j.

Original entry on oeis.org

1, 3, 2, 9, 12, 4, 27, 54, 36, 8, 81, 216, 216, 96, 16, 243, 810, 1080, 720, 240, 32, 729, 2916, 4860, 4320, 2160, 576, 64, 2187, 10206, 20412, 22680, 15120, 6048, 1344, 128, 6561, 34992, 81648, 108864, 90720, 48384, 16128, 3072, 256

Offset: 0

N. J. A. Sloane

Row sums give A000351; central terms give A119309. - Reinhard Zumkeller, May 14 2006

Triangle of coefficients in expansion of (3 + 2x)^n, where n is a nonnegative integer. - Zagros Lalo, Jul 23 2018

			Triangle begins:
   1;
   3,   2;
   9,  12,   4;
  27,  54,  36,   8;
  81, 216, 216,  96,  16;
  ...

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 44, 48

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A013620, A000079, A000244, A013613, A038221.

a038220 n k = a038220_tabl !! n !! k
a038220_row n = a038220_tabl !! n
a038220_tabl = iterate (\row ->
   zipWith (+) (map (* 3) (row ++ [0])) (map (* 2) ([0] ++ row))) [1]
-- Reinhard Zumkeller, May 26 2013, Apr 02 2011

t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 3 t[n - 1, k] + 2 t[n - 1, k - 1]]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Zagros Lalo, Jul 23 2018 *)
Table[CoefficientList[ Expand[(3 + 2x)^n], x], {n, 0, 9}] // Flatten  (* Zagros Lalo, Jul 23 2018 *)
Table[CoefficientList[Binomial[i, j] *3^(i - j)*2^j, x], {i, 0, 9}, {j, 0, i}] // Flatten (* Zagros Lalo, Jul 23 2018 *)

T(i,j)=binomial(i,j)*3^(i-j)*2^j \\ Charles R Greathouse IV, Jul 19 2016

T(n,k) = A007318(n,k) * A036561(n,k). - Reinhard Zumkeller, May 14 2006
G.f.: 1/(1 - 3*x - 2*x*y). - Ilya Gutkovskiy, Apr 21 2017
T(0,0) = 1; T(n,k) = 3 T(n-1,k) + 2 T(n-1,k-1) for k = 0...n; T(n,k)=0 for n or k < 0. - Zagros Lalo, Jul 23 2018

A304252 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 6*T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 1, 1, 6, 1, 12, 1, 18, 36, 1, 24, 108, 1, 30, 216, 216, 1, 36, 360, 864, 1, 42, 540, 2160, 1296, 1, 48, 756, 4320, 6480, 1, 54, 1008, 7560, 19440, 7776, 1, 60, 1296, 12096, 45360, 46656, 1, 66, 1620, 18144, 90720, 163296, 46656, 1, 72, 1980, 25920, 163296, 435456, 326592, 1, 78, 2376, 35640

Offset: 0

Zagros Lalo, May 09 2018

The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A013613 ((1+6*x)^n).

The coefficients in the expansion of 1/(1-x-6*x^2) are given by the sequence generated by the row sums.

			Triangle begins:
  1;
  1;
  1,  6;
  1, 12;
  1, 18,   36;
  1, 24,  108;
  1, 30,  216,   216;
  1, 36,  360,   864;
  1, 42,  540,  2160,   1296;
  1, 48,  756,  4320,   6480;
  1, 54, 1008,  7560,  19440,    7776;
  1, 60, 1296, 12096,  45360,   46656;
  1, 66, 1620, 18144,  90720,  163296,   46656;
  1, 72, 1980, 25920, 163296,  435456,  326592;
  1, 78, 2376, 35640, 272160,  979776, 1306368,  279936;
  1, 84, 2808, 47520, 427680, 1959552, 3919104, 2239488;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72.

Zagros Lalo, Left-justified triangle
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1+6x)^n

Row sums give A015441.

Cf. A013613.

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, t[n - 1, k] + 6 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten (* Robert G. Wilson v, May 19 2018 *)
Table[6^k Binomial[n - k, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten

T(n,k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, T(n-1,k) + 6*T(n-2,k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 10 2018

T(n,k) = 6^k*binomial(n-k,k), n >= 0, 0 <= k <= floor(n/2).

A304255 Triangle read by rows: T(0,0) = 1; T(n,k) = 6*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 6, 36, 1, 216, 12, 1296, 108, 1, 7776, 864, 18, 46656, 6480, 216, 1, 279936, 46656, 2160, 24, 1679616, 326592, 19440, 360, 1, 10077696, 2239488, 163296, 4320, 30, 60466176, 15116544, 1306368, 45360, 540, 1, 362797056, 100776960, 10077696, 435456, 7560, 36

Offset: 0

Zagros Lalo, May 09 2018

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013613 ((1+6*x)^n).
The coefficients in the expansion of 1/(1-6x-x^2) are given by the sequence generated by the row sums.
The row sums are Denominators of continued fraction convergent to sqrt(10), see A005668.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 6.162277660..., a metallic mean (see A176398), when n approaches infinity.

			Triangle begins:
1;
6;
36, 1;
216, 12;
1296, 108, 1;
7776, 864, 18;
46656, 6480, 216, 1;
279936, 46656, 2160, 24;
1679616, 326592, 19440, 360, 1;
10077696, 2239488, 163296, 4320, 30;
60466176, 15116544, 1306368, 45360, 540, 1;
362797056, 100776960, 10077696, 435456, 7560, 36;
2176782336, 665127936, 75582720, 3919104, 90720, 756, 1;
13060694016, 4353564672, 554273280, 33592320, 979776, 12096, 42;
78364164096, 28298170368, 3990767616, 277136640, 9797760, 163296, 1008, 1;
470184984576, 182849716224, 28298170368, 2217093120, 92378880, 1959552, 18144, 48;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 94.

Zagros Lalo, Left-justified triangle
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1+6x)^n

Row sums give A005668.
Cf. A000400 (column 0), A053469 (column 1), A081136 (column 2), A081144 (column 3).
Cf. A013613.
Cf. A176398.

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 6 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten

T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, 6*T(n-1, k) + T(n-2, k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 26 2018

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A081580 Pascal-(1,5,1) array.

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A038255 Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).

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A013620 Triangle of coefficients in expansion of (2+3x)^n.

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A038220 Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.

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A304252 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 6*T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

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A304255 Triangle read by rows: T(0,0) = 1; T(n,k) = 6*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

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A038220 Triangle whose (i,j)-th entry is binomial(i,j)3^(i-j)2^j.