A218017 -id:A218017 - cp's OEIS frontend

A217629 Triangle, read by rows, where T(n,k) = k!C(n, k)3^(n-k) for n>=0, k=0..n.

1, 3, 1, 9, 6, 2, 27, 27, 18, 6, 81, 108, 108, 72, 24, 243, 405, 540, 540, 360, 120, 729, 1458, 2430, 3240, 3240, 2160, 720, 2187, 5103, 10206, 17010, 22680, 22680, 15120, 5040, 6561, 17496, 40824, 81648, 136080, 181440, 181440, 120960, 40320

Offset: 0

Vincenzo Librandi, Nov 10 2012

Triangle formed by the derivatives of x^n evaluated at x=3.
Sum(T(n,k), k=0..n) = A053486(n) (see the Formula section of A053486). Also:
first column:     A000244;
second column:    A027471;
third column:   2*A027472;
fourth column:  6*A036216;
fifth column:  24*A036217.

			Triangle begins:
1;
3,     1;
9,     6,     2;
27,    27,    18,     6;
81,    108,   108,    72,     24;
243,   405,   540,    540,    360,    120;
729,   1458,  2430,   3240,   3240,   2160,    720;
2187,  5103,  10206,  17010,  22680,  22680,   15120,   5040;
6561,  17496, 40824,  81648,  136080, 181440,  181440,  120960,  40320; etc.

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A000244, A027471, A027472, A036216, A036217, A053486, A090802, A218016, A218017.

[Factorial(n)/Factorial(n-k)*3^(n-k): k in [0..n], n in [0..10]];

Flatten[Table[n!/(n-k)!*3^(n-k), {n, 0, 10}, {k, 0, n}]]

T(n,k) = 3^(n-k)*n!/(n-k)! for n>=0, k=0..n.

E.g.f. (by columns): exp(3x)*x^k.

A218016 Triangle, read by rows, where T(n,k) = k!C(n, k)5^(n-k) for n>=0, k=0..n.

Original entry on oeis.org

1, 5, 1, 25, 10, 2, 125, 75, 30, 6, 625, 500, 300, 120, 24, 3125, 3125, 2500, 1500, 600, 120, 15625, 18750, 18750, 15000, 9000, 3600, 720, 78125, 109375, 131250, 131250, 105000, 63000, 25200, 5040, 390625, 625000, 875000, 1050000, 1050000, 840000, 504000, 201600, 40320

Offset: 0

Vincenzo Librandi, Nov 10 2012

Triangle formed by the derivatives of x^n evaluated at x=5.
Sum(T(n,k), k=0..n) = A080954(n) (see the Formula section of A080954). . Also:
first column:    A000351;
second column:   A053464;
third column:  2*A084902;
fourth column: 6*A081143.

			Triangle begins:
1;
5,      1;
25,     10,     2;
125,    75,     30,     6;
625,    500,    300,    120,     24;
3125,   3125,   2500,   1500,    600,     120;
15625,  18750,  18750,  15000,   9000,    3600,   720;
78125,  109375, 131250, 131250,  105000,  63000,  25200,  5040;
390625, 625000, 875000, 1050000, 1050000, 840000, 504000, 201600, 40320; etc.

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A000351, A053464, A080954, A081143, A084902, A090802, A217629, A218017.

[Factorial(n)/Factorial(n-k)*5^(n-k): k in [0..n], n in [0..10]];

Flatten[Table[n!/(n-k)!*5^(n-k), {n, 0, 10}, {k, 0, n}]]

T(n,k) = 5^(n-k)*n!/(n-k)! for n>=0, k=0..n.

E.g.f. (by columns): exp(5x)*x^k.

A218018 Triangle, read by rows, where T(n,k) = k!C(n, k)11^(n-k) for n>=0, k=0..n.

Original entry on oeis.org

1, 11, 1, 121, 22, 2, 1331, 363, 66, 6, 14641, 5324, 1452, 264, 24, 161051, 73205, 26620, 7260, 1320, 120, 1771561, 966306, 439230, 159720, 43560, 7920, 720, 19487171, 12400927, 6764142, 3074610, 1118040, 304920, 55440, 5040, 214358881

Offset: 0

Vincenzo Librandi, Nov 17 2012

Triangle formed by the derivatives of x^n evaluated at x=11. Also:
first column:   A001020;
second column:  A081127;
third column: 2*A081141.

			Triangle begins:
1;
11,       1;
121,      22,       2;
1331,     363,      66,      6;
14641,    5324,     1452,    264,     24;
161051,   73205,    26620,   7260,    1320,    120;
1771561,  966306,   439230,  159720,  43560,   7920,   720;
19487171, 12400927, 6764142, 3074610, 1118040, 304920, 55440, 5040; etc.

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A001020, A081127, A081141, A217629, A218016, A218017.

[Factorial(n)/Factorial(n-k)*11^(n-k): k in [0..n], n in [0..10]];

Flatten[Table[n!/(n-k)! * 11^(n-k), {n, 0, 10}, {k, 0, n}]]

T(n,k) = 11^(n-k)*n!/(n-k)! for n>=0, k=0..n.

E.g.f. (by columns): exp(11*x)*x^k.

cp's OEIS Frontend

A217629 Triangle, read by rows, where T(n,k) = k!C(n, k)3^(n-k) for n>=0, k=0..n.

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A218016 Triangle, read by rows, where T(n,k) = k!C(n, k)5^(n-k) for n>=0, k=0..n.

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Formula

A218018 Triangle, read by rows, where T(n,k) = k!C(n, k)11^(n-k) for n>=0, k=0..n.

Views

Author

Keywords

Comments

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Links

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Magma

Mathematica

Formula

cp's OEIS Frontend

A217629 Triangle, read by rows, where T(n,k) = k!*C(n, k)*3^(n-k) for n>=0, k=0..n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A218016 Triangle, read by rows, where T(n,k) = k!*C(n, k)*5^(n-k) for n>=0, k=0..n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A218018 Triangle, read by rows, where T(n,k) = k!*C(n, k)*11^(n-k) for n>=0, k=0..n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A217629 Triangle, read by rows, where T(n,k) = k!C(n, k)3^(n-k) for n>=0, k=0..n.

A218016 Triangle, read by rows, where T(n,k) = k!C(n, k)5^(n-k) for n>=0, k=0..n.

A218018 Triangle, read by rows, where T(n,k) = k!C(n, k)11^(n-k) for n>=0, k=0..n.