A178987 -id:A178987 - cp's OEIS frontend

A127276 Hankel transform of A127275.

1, 1, -2, -16, -64, -208, -608, -1664, -4352, -11008, -27136, -65536, -155648, -364544, -843776, -1933312, -4390912, -9895936, -22151168, -49283072, -109051904, -240123904, -526385152, -1149239296, -2499805184, -5419040768

Offset: 0

Paul Barry, Jan 10 2007

sign

The inverse binomial transform of this sequence yields 1, 0, -3, -8,..., which is 1 followed by the negated terms of A005563. [Paul Curtz, Dec 07 2010]

The smallest odd prime divisor of a(n) is >= 13. - Vladimir Shevelev, Feb 03 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A076616, A176126, A236999.

[2^n-n*(n+1)*2^(n-2): n in [0..30]]; // Vincenzo Librandi, Aug 11 2011

A127276:=n->2^n - n*(n + 1)*2^(n - 2); seq(A127276(n), n=0..26); # Wesley Ivan Hurt, Dec 02 2013

Table[2^n - n*(n + 1)*2^(n - 2), {n, 0, 26}] (* Wesley Ivan Hurt, Dec 02 2013 *)

Conjecture: G.f.: -(4*x-1)*(x-1) / ( (2*x-1)^3 ) and a(n) = 2^n-n*(n+1)*2^(n-2). - R. J. Mathar, Dec 11 2010

a(n) = A178987(n) - A178987(n+1). - Klaus Brockhaus, Jan 08 2011

A383150 a(n) = Sum_{k=0..n} k^3 * (-1)^k * 3^(n-k) * binomial(n,k).

Original entry on oeis.org

0, -1, 2, 18, 64, 160, 288, 224, -1024, -6912, -28160, -95744, -294912, -851968, -2351104, -6266880, -16252928, -41222144, -102629376, -251527168, -608174080, -1453326336, -3437232128, -8055160832, -18723373056, -43201331200, -99019128832, -225586446336

Offset: 0

Seiichi Manyama, Apr 18 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..700

Cf. A001787, A178987, A383151, A383152, A383155.

[2^(n-3) * (-12*n+9*n^2-n^3): n in [0..30]]; // Vincenzo Librandi, May 02 2025

Table[2^(n-3)*(-12*n+9*n^2-n^3),{n,0,30}] (* Vincenzo Librandi, May 02 2025 *)

PARI
```
a(n) = 2^(n-3)*(-12*n+9*n^2-n^3);
```

a(n) = 2^(n-3) * (-12*n + 9*n^2 - n^3).

A383151 a(n) = Sum_{k=0..n} k^4 * (-1)^k * 3^(n-k) * binomial(n,k).

Original entry on oeis.org

0, -1, 10, 36, 40, -160, -1152, -4480, -13568, -34560, -74240, -123904, -92160, 425984, 2867200, 11796480, 40763392, 128122880, 378667008, 1070858240, 2928148480, 7795113984, 20300431360, 51900317696, 130610626560, 324219699200, 795206483968, 1929715384320

Offset: 0

Seiichi Manyama, Apr 18 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..700

Cf. A001787, A178987, A383150, A383152, A383155.

[&+[k^4 * (-1)^k * 3^(n-k) * Binomial(n,k): k in [0..n]]: n in [0..29]]; // Vincenzo Librandi, Apr 23 2025

Table[Sum[(k^4*(-1)^k*3^(n-k))*Binomial[n,k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 23 2025 *)

a(n) = 2^(n-4)*(-66*n+75*n^2-18*n^3+n^4);

a(n) = 2^(n-4) * (-66*n + 75*n^2 - 18*n^3 + n^4).

A383152 a(n) = Sum_{k=0..n} k^5 * (-1)^k * 3^(n-k) * binomial(n,k).

Original entry on oeis.org

0, -1, 26, 18, -272, -1400, -4032, -7168, -1024, 55296, 294400, 1086976, 3354624, 9132032, 22249472, 47923200, 85983232, 99155968, -102629376, -1237712896, -5688524800, -20775960576, -67868033024, -207022456832, -602167836672, -1690304512000, -4613767954432

Offset: 0

Seiichi Manyama, Apr 18 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..700

Cf. A001787, A178987, A383150, A383151, A383155.

[&+[k^5 * (-1)^k * 3^(n-k) * Binomial(n, k): k in [0..n]]: n in [0..29]]; // Vincenzo Librandi, Apr 23 2025

Table[2^(n-5)*(-480*n+690*n^2-255*n^3+30*n^4-n^5),{n,0,50}] (* Vincenzo Librandi, Apr 24 2025 *)

a(n) = 2^(n-5)*(-480*n+690*n^2-255*n^3+30*n^4-n^5);

a(n) = 2^(n-5) * (-480*n + 690*n^2 - 255*n^3 + 30*n^4 - n^5).

A383155 a(n) = Sum_{k=0..n} k^6 * (-1)^k * 3^(n-k) * binomial(n,k).

Original entry on oeis.org

0, -1, 58, -180, -1304, -2920, 1008, 34496, 163840, 525312, 1285120, 2241536, 1124352, -12113920, -72052736, -282378240, -924581888, -2699493376, -7201751040, -17666670592, -39507722240, -77918109696, -121883328512, -78622228480, 453588811776, 2904974950400, 11885785120768

Offset: 0

Seiichi Manyama, Apr 18 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..700

Cf. A001787, A178987, A383150, A383151, A383152.

[&+[k^6 * (-1)^k * 3^(n-k) * Binomial(n,k): k in [0..n]]: n in [0..29]]; // Vincenzo Librandi, Apr 23 2025

Table[Sum[(k^6*(-1)^k*3^(n-k))*Binomial[n,k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 23 2025 *)

a(n) = 2^(n-6)*(-4368*n+7290*n^2-3555*n^3+645*n^4-45*n^5+n^6);

a(n) = 2^(n-6) * (-4368*n + 7290*n^2 - 3555*n^3 + 645*n^4 - 45*n^5 + n^6).

A181407 a(n) = (n-4)(n-3)2^(n-2).

Original entry on oeis.org

3, 3, 2, 0, 0, 16, 96, 384, 1280, 3840, 10752, 28672, 73728, 184320, 450560, 1081344, 2555904, 5963776, 13762560, 31457280, 71303168, 160432128, 358612992, 796917760, 1761607680, 3875536896, 8489271296, 18522046464, 40265318400, 87241523200, 188441690112

Offset: 0

Paul Curtz, Jan 28 2011

Binomial transform of (3, 0, -1, followed by A005563).
The sequence and its successive differences are:
   3,  3,  2,   0,   0,   16,   96,  384,      a(n),
   0, -1, -2,   0,  16,   80,  288,  896,      A178987,
  -1, -1,  2,  16,  64,  208,  608, 2688,     -A127276,
   0,  3, 14,  48, 144,  400, 1056, 2688,      A176027,
   3, 11, 34,  96, 256,  656, 1632, 3968,      A084266(n+1)
   8, 23, 62, 160, 400,  976, 2336, 5504,
  15, 39, 98, 240, 576, 1360, 3168, 7296.
Division of the k-th column by abs(A174882(k)) gives
   3,  3,  1,  0,  0,  1,  3,  3,  5,  15,  21, 14,   A064038(n-3),
   0, -1, -1,  0,  1,  5,  9,  7, 10,  27,  35, 22,   A160050(n-3),
  -1, -1,  1,  2,  4, 13, 19, 13, 17,  43,  53, 32,   A176126,
   0,  3,  7,  6,  9, 25, 33, 21, 26,  63,  75, 44,   A178242,
   3, 11, 17, 12, 16, 41, 51, 31, 37,  87, 101, 58,
   8  23, 31, 20, 25, 61, 73, 43, 50, 115, 131, 74,
  15, 39, 49, 30, 36, 85, 99, 57, 65, 147, 165, 92.
Columns are (or from) A005563, A142463, A056220, A002378, A000290, A001844, A058331, A002061, A002522, A097080, A093328, A014206.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A176027, A181318.

List([0..40], n-> (n-4)*(n-3)*2^(n-2)); # G. C. Greubel, Feb 21 2019

[(n-4)*(n-3)*2^(n-2): n in [0..40] ]; // Vincenzo Librandi, Feb 01 2011

Table[(n-4)*(n-3)*2^(n-2), {n,0,40}] (* G. C. Greubel, Feb 21 2019 *)

vector(40, n, n--; (n-4)*(n-3)*2^(n-2)) \\ G. C. Greubel, Feb 21 2019

[(n-4)*(n-3)*2^(n-2) for n in (0..40)] # G. C. Greubel, Feb 21 2019

a(n) = 16*A001788(n-4).
a(n+1) - a(n) = A178987(n).
G.f.: (3 - 15*x + 20*x^2) / (1-2*x)^3. - R. J. Mathar, Jan 30 2011
E.g.f.: (x^2 - 3*x + 3)*exp(2*x). - G. C. Greubel, Feb 21 2019

A383149 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = (-1)^k * [m^k] (1/2^(m-n)) * Sum_{k=0..m} k^n * (-1)^m * 3^(m-k) * binomial(m,k).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 12, 9, 1, 0, 66, 75, 18, 1, 0, 480, 690, 255, 30, 1, 0, 4368, 7290, 3555, 645, 45, 1, 0, 47712, 88536, 52290, 12705, 1365, 63, 1, 0, 608016, 1223628, 831684, 249585, 36120, 2562, 84, 1, 0, 8855040, 19019664, 14405580, 5073012, 915705, 87696, 4410, 108, 1

Offset: 0

Seiichi Manyama, Apr 18 2025

			f_0(m) = 1.
f_1(m) =      -m.
f_2(m) =    -3*m +     m^2.
f_3(m) =   -12*m +   9*m^2 -     m^3.
f_4(m) =   -66*m +  75*m^2 -  18*m^3 +    m^4.
f_5(m) =  -480*m + 690*m^2 - 255*m^3 + 30*m^4 - m^5.
Triangle begins:
  1;
  0,     1;
  0,     3,     1;
  0,    12,     9,     1;
  0,    66,    75,    18,     1;
  0,   480,   690,   255,    30,    1;
  0,  4368,  7290,  3555,   645,   45,  1;
  0, 47712, 88536, 52290, 12705, 1365, 63, 1;
  ...

Eric Weisstein's World of Mathematics, Rising Factorial

Columns k=0..3 give A000007, A123227(n-1), A383163, A383164.
Row sums give A122704.
Cf. A001787, A178987, A383150, A383151, A383152, A383155.
Cf. A129062, A209849, A383140.

T(n, k) = sum(j=k, n, 2^(n-j)*stirling(n, j, 2)*abs(stirling(j, k, 1)));

def a_row(n):
    s = sum(2^(n-k)*stirling_number2(n, k)*rising_factorial(x, k) for k in (0..n))
    return expand(s).list()
for n in (0..9): print(a_row(n))

f_n(m) = (1/2^(m-n)) * Sum_{k=0..m} k^n * (-1)^m * 3^(m-k) * binomial(m,k).
T(n,k) = [m^k] f_n(-m).
T(n,k) = Sum_{j=k..n} 2^(n-j) * Stirling2(n,j) * |Stirling1(j,k)|.
T(n,k) = [x^k] Sum_{k=0..n} 2^(n-k) * Stirling2(n,k) * RisingFactorial(x,k).
Sum_{k=0..n} (-1)^k * T(n,k) = f_m(1) = -2^(n-1) for n > 0.
E.g.f. of column k (with leading zeros): g(x)^k / k! with g(x) = -log(1 - (exp(2*x) - 1)/2).

cp's OEIS Frontend

A127276 Hankel transform of A127275.

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A383150 a(n) = Sum_{k=0..n} k^3 * (-1)^k * 3^(n-k) * binomial(n,k).

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A383151 a(n) = Sum_{k=0..n} k^4 * (-1)^k * 3^(n-k) * binomial(n,k).

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A383152 a(n) = Sum_{k=0..n} k^5 * (-1)^k * 3^(n-k) * binomial(n,k).

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A383155 a(n) = Sum_{k=0..n} k^6 * (-1)^k * 3^(n-k) * binomial(n,k).

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A181407 a(n) = (n-4)*(n-3)*2^(n-2).

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A383149 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = (-1)^k * [m^k] (1/2^(m-n)) * Sum_{k=0..m} k^n * (-1)^m * 3^(m-k) * binomial(m,k).

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A181407 a(n) = (n-4)(n-3)2^(n-2).