A171531 -id:A171531 - cp's OEIS frontend

A171631 Triangle read by rows: T(n,k) = n(binomial(n-2, k-1) + nbinomial(n-2, k)), n > 0 and 0 <= k <= n - 1.

1, 4, 2, 9, 12, 3, 16, 36, 24, 4, 25, 80, 90, 40, 5, 36, 150, 240, 180, 60, 6, 49, 252, 525, 560, 315, 84, 7, 64, 392, 1008, 1400, 1120, 504, 112, 8, 81, 576, 1764, 3024, 3150, 2016, 756, 144, 9, 100, 810, 2880, 5880, 7560, 6300, 3360, 1080, 180, 10, 121, 1100

Offset: 1

Roger L. Bagula, Dec 13 2009

If T(0,0) = 0 is prepended, then row sums give A001788.

			Triangle begins:
n\k|  0    1     2     3     4     6    7    8  9
-------------------------------------------------
1  |  1
2  |  4    2
3  |  9   12     3
4  | 16   36    24     4
5  | 25   80    90    40     5
6  | 36  150   240   180    60     6
7  | 49  252   525   560   315    84    7
8  | 64  392  1008  1400  1120   504  112    8
9  | 81  576  1764  3024  3150  2016  756  144  9
... reformatted. - _Franck Maminirina Ramaharo_, Oct 02 2018

Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Publications, 1945, p. 32.

Cf. A003506, A007318, A127952, A171531.

Table[CoefficientList[n*(x + n)*(x + 1)^(n - 2), x], {n, 1, 12}]//Flatten

T(n, k) := n*(binomial(n - 2, k - 1) + n*binomial(n - 2, k))$
tabl(nn) := for n:1 thru nn do print(makelist(T(n, k), k, 0, n - 1))$ /* Franck Maminirina Ramaharo, Oct 02 2018 */

Let p(x;n) = (x + 1)^n. Then row n are the coefficients in the expansion of p''(x;n) - x*p'(x;n) + n*p(x;n) = n*(x + n)*(x + 1)^(n - 2).
From Franck Maminirina Ramaharo, Oct 02 2018: (Start)
T(n,1) = A000290(n).
T(n,2) = A011379(n).
T(n,3) = 3*A002417(n-2).
T(n,n-2) = A046092(n-1).
T(n,n-3) = 9*A000292(n-2).
G.f.: y*(x*y - y - 1)/(x*y + y - 1)^3. (End)

Edited and new name by Franck Maminirina Ramaharo, Oct 02 2018

A171633 Coefficients of a Hermite-like polynomial from Eulerian polynomials: p(x,n) = Sum_{k=1..n+1} [Eulerian(n + 1, k - 1)x^(k - 1)]; q(x,n) = p''(x,n) - xp'(x,n) + n*p(x,n).

Original entry on oeis.org

1, 4, 4, 25, 28, 11, 136, 234, 144, 26, 609, 2040, 1590, 624, 57, 2388, 15096, 19056, 9648, 2412, 120, 8593, 95196, 208893, 148336, 54267, 8628, 247, 29224, 532918, 1961928, 2205850, 1063000, 285786, 29272, 502, 95689, 2739256, 16059128

Offset: 1

Roger L. Bagula, Dec 13 2009

Row sums are {1, 8, 64, 540, 4920, 48720, 524160, 6108480, 76809600, 1037836800, 15008716800, 231437606400, ...}.

The important observation here is that the modulo two pattern is the same as the Hermite product A171531 type polynomials.

			{1},
{4, 4},
{25, 28, 11},
{136, 234, 144, 26},
{609, 2040, 1590, 624, 57},
{2388, 15096, 19056, 9648, 2412, 120},
{8593, 95196, 208893, 148336, 54267, 8628, 247},
{29224, 532918, 1961928, 2205850, 1063000, 285786, 29272, 502},
{95689, 2739256, 16059128, 28938232, 20207530, 7250696, 1422304, 95752, 1013},
{305284, 13239252, 118078464, 329909376, 350572104, 171167736, 47500128, 6757056, 305364, 2036}

Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Book, New York, 1945, page 32.

t[n_, k_] := Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]
p[x_, n_] := Sum[t[n + 1, k - 1]*x^(k - 1), {k, 1, n + 1}]
b = Table[CoefficientList[D[p[x, n], {x, 2}] - x*D[p[x, n], {x, 1}] + n*p[x, n], x], {n, 1, 10}]
Flatten[%]

p(x,n) = p(x,n) = Sum_{k=1..n+1} [Eulerian(n + 1, k - 1)*x^(k - 1), ];

q(x,n) = p''(x,n) - x*p'(x,n) + n*p(x,n).

cp's OEIS Frontend

A171631 Triangle read by rows: T(n,k) = n(binomial(n-2, k-1) + nbinomial(n-2, k)), n > 0 and 0 <= k <= n - 1.

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A171633 Coefficients of a Hermite-like polynomial from Eulerian polynomials: p(x,n) = Sum_{k=1..n+1} [Eulerian(n + 1, k - 1)x^(k - 1)]; q(x,n) = p''(x,n) - xp'(x,n) + n*p(x,n).

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cp's OEIS Frontend

A171631 Triangle read by rows: T(n,k) = n*(binomial(n-2, k-1) + n*binomial(n-2, k)), n > 0 and 0 <= k <= n - 1.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Maxima

Formula

Extensions

A171633 Coefficients of a Hermite-like polynomial from Eulerian polynomials: p(x,n) = Sum_{k=1..n+1} [Eulerian(n + 1, k - 1)*x^(k - 1)]; q(x,n) = p''(x,n) - x*p'(x,n) + n*p(x,n).

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Author

Keywords

Comments

Examples

References

Programs

Mathematica

Formula

A171631 Triangle read by rows: T(n,k) = n(binomial(n-2, k-1) + nbinomial(n-2, k)), n > 0 and 0 <= k <= n - 1.

A171633 Coefficients of a Hermite-like polynomial from Eulerian polynomials: p(x,n) = Sum_{k=1..n+1} [Eulerian(n + 1, k - 1)x^(k - 1)]; q(x,n) = p''(x,n) - xp'(x,n) + n*p(x,n).