A157011 -id:A157011 - cp's OEIS frontend

A157012 Riordan's general Eulerian recursion: T(n,k) = (k+2)T(n-1, k) + (n-k) T(n-1, k-1), with T(n,0) = 1, T(n,n) = 0.

1, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 18, 14, 1, 0, 1, 58, 110, 33, 1, 0, 1, 179, 672, 495, 72, 1, 0, 1, 543, 3583, 5163, 1917, 151, 1, 0, 1, 1636, 17590, 43730, 32154, 6808, 310, 1, 0, 1, 4916, 81812, 324190, 411574, 176272, 22904, 629, 1, 0

Offset: 0

Roger L. Bagula, Feb 21 2009

Row sums are:
{1, 1, 2, 7, 34, 203, 1420, 11359, 102230, 1022299,...}.
This recursion set doesn't seem to produce the Eulerian 2nd A008517.
The Mathematica code gives ten sequences of which the first few are in the OEIS (see Crossrefs section). - G. C. Greubel, Feb 22 2019

			Triangle begins with:
1.
1,    0.
1,    1,     0.
1,    5,     1,      0.
1,   18,    14,      1,      0.
1,   58,   110,     33,      1,      0.
1,  179,   672,    495,     72,      1,     0.
1,  543,  3583,   5163,   1917,    151,     1,   0.
1, 1636, 17590,  43730,  32154,   6808,   310,   1,   0.
1, 4916, 81812, 324190, 411574, 176272, 22904, 629,   1,   0.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 214-215

G. C. Greubel, Rows n=0..100 of triangle, flattened

Cf. A008517.

Cf. A157011 (m=0), A008292 (m=1), This Sequence (m=2), A157013 (m=3).

e[n_, 0, m_]:= 1;
e[n_, k_, m_]:= 0 /; k >= n;
e[n_, k_, m_]:= (k+m)*e[n-1, k, m] +(n-k+1-m)*e[n-1, k-1, m];
Table[Flatten[Table[Table[e[n, k, m], {k,0,n-1}], {n,1,10}]], {m,0,10}]
T[n_, 0]:= 1; T[n_, n_]:= 0; T[n_, k_]:= T[n, k] = (k+2)*T[n-1, k] +(n-k) *T[n-1, k-1]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 22 2019 *)

{T(n, k) = if(k==0, 1, if(k==n, 0, (k+2)*T(n-1, k) + (n-k)* T(n-1, k-1)))};
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Feb 22 2019

def T(n, k):
    if (k==0): return 1
    elif (k==n): return 0
    else: return (k+2)*T(n-1, k) + (n-k)* T(n-1, k-1)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 22 2019

e(n,k,m) = (k+m)*e(n-1, k, m) + (n-k+1-m)*e(n-1, k-1, m) for m=2.

T(n,k) = (k+2)*T(n-1, k) + (n-k)*T(n-1, k-1), with T(n,0) = 1, T(n,n) = 0. - G. C. Greubel, Feb 22 2019

A157013 Riordan's general Eulerian recursion: T(n, k) = (k+2)T(n-1, k) + (n-k-1) T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-1)^(n-1).

Original entry on oeis.org

1, 1, -1, 1, -4, 1, 1, -15, 5, -1, 1, -58, 10, -6, 1, 1, -229, -66, -26, 7, -1, 1, -912, -1017, -288, 23, -8, 1, 1, -3643, -8733, -4779, -415, -41, 9, -1, 1, -14566, -61880, -63606, -17242, -1158, 40, -10, 1, 1, -58257, -396796, -691036, -375118, -60990, -1956, -60, 11, -1

Offset: 1

Roger L. Bagula, Feb 21 2009

Row sums are {1, 0, -2, -10, -52, -314, -2200, -17602, -158420, -1584202, ...}.
This recursion set doesn't seem to produce the Eulerian 2nd A008517.
The Mathematica code gives ten sequences of which the first few are in the OEIS (see Crossrefs section). - G. C. Greubel, Feb 22 2019

			Triangle begins with:
1.
1,     -1.
1,     -4,       1.
1,    -15,       5,      -1.
1,    -58,      10,      -6,       1.
1,   -229,     -66,     -26,       7,     -1.
1,   -912,   -1017,    -288,      23,     -8,     1.
1,  -3643,   -8733,   -4779,    -415,    -41,     9,   -1.
1, -14566,  -61880,  -63606,  -17242,  -1158,    40,  -10,   1.
1, -58257, -396796, -691036, -375118, -60990, -1956,  -60,  11,  -1.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 214-215

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A008517.

Cf. A157011 (m=0), A008292 (m=1), A157012 (m=2), this sequence (m=3).

e[n_, 0, m_]:= 1;
e[n_, k_, m_]:= 0 /; k >= n;
e[n_, k_, m_]:= (k+m)*e[n-1, k, m] + (n-k+1-m)*e[n-1, k-1, m];
Table[Flatten[Table[Table[e[n, k, m], {k,0,n-1}], {n,1,10}]], {m,0,10}]
T[n_,1]:=1; T[n_,n_]:=(-1)^(n-1); T[n_,k_]:= T[n,k] = (k+2)*T[n-1,k] + (n-k-1)*T[n-1,k-1]; Table[T[n,k], {n,1,10}, {k,1,n}]//Flatten (* G. C. Greubel, Feb 22 2019 *)

{T(n, k) = if(k==1, 1, if(k==n, (-1)^(n-1), (k+2)*T(n-1, k) + (n-k-1)* T(n-1, k-1)))};
for(n=1, 10, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Feb 22 2019

def T(n, k):
    if (k==1): return 1
    elif (k==n): return (-1)^(n-1)
    else: return (k+2)*T(n-1, k) + (n-k-1)* T(n-1, k-1)
[[T(n, k) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Feb 22 2019

e(n,k,m)= (k+m)*e(n-1, k, m) + (n-k+1-m)*e(n-1, k-1, m) with m=3.

T(n, k) = (k+2)*T(n-1, k) + (n-k-1)*T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-1)^(n-1). - G. C. Greubel, Feb 22 2019

A180246 Triangle T(n,k) read by rows: T(n,k) = Sum_{j=0..k} (-1)^j binomial(n+1,j)(k+2-j)^n, 0 <= k < n.

Original entry on oeis.org

2, 4, -3, 8, -5, 4, 16, 1, 11, -5, 32, 51, 46, -14, 6, 64, 281, 337, 22, 22, -7, 128, 1163, 2472, 1121, 176, -27, 8, 256, 4257, 15703, 15493, 4419, 163, 37, -9, 512, 14563, 88354, 155980, 88486, 14398, 622, -44, 10, 1024, 47785, 455357, 1310024, 1310816, 454730, 48170, 848, 56, -11

Offset: 1

Roger L. Bagula, Aug 19 2010

Row sums are apparently the 2nd column of A156984, 2, 1, 7, 23,...

Generalizes A008292 in the sense that a term "2" is added to the factor that is raised to the n-th power in the sum of the definition. A term "1" would generate A008292, too (up to index shifts).

			Triangle begins with:
     2;
     4,    -3;
     8,    -5,      4;
    16,     1,     11,      -5;
    32,    51,     46,     -14,       6;
    64,   281,    337,      22,      22,     -7;
   128,  1163,   2472,    1121,     176,    -27,     8;
   256,  4257,  15703,   15493,    4419,    163,    37,  -9;
   512, 14563,  88354,  155980,   88486,  14398,   622, -44, 10;
  1024, 47785, 455357, 1310024, 1310816, 454730, 48170, 848, 56, -11;
  ...

B. Harris and C J. Park, A generalization of Eulerian numbers with a probabilistic Application, Statistics and Probability Letters 20 (1994), page 40

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A008292, A157011.

Flat(List([1..12], n-> List([0..n-1], k-> Sum([0..k], j-> (-1)^j*Binomial(n+1, j)*(k-j+2)^n )))); # G. C. Greubel, Feb 23 2019

[[(&+[(-1)^j*Binomial(n+1, j)*(k-j+2)^n: j in [0..k]]): k in [0..n-1]]: n in [1..12]]; // G. C. Greubel, Feb 23 2019

A180246 := proc(n,k) add( (-1)^v*binomial(n+1,v)*(k+2-v)^n,v=0..k) ; end proc: # R. J. Mathar, Jan 29 2011
P := proc(n,x) option remember; if n = 0 then 1 else
  (n*x+2*(1-x))*P(n-1,x)+x*(1-x)*diff(P(n-1,x),x);
  expand(%) fi end:
A180246 := (n,k) -> coeff(P(n,x),x,k):
seq(print(seq(A180246(n,k),k=0..n-1)),n=0..10);  # Peter Luschny, Mar 07 2014

t[n_, j_, d_]:= Sum[(-1)^v *Binomial[n+1, v](j+d-v)^n, {v, 0, j}];
Table[Flatten[Table[Table[t[n,k,m], {k,0,n-1}], {n,1,10}]], {m,0,10}]
(* This sequence corresponds to m=2 *)
Table[Sum[(-1)^j*Binomial[n+1, j]*(k-j+2)^n, {j,0,k}], {n,1,12}, {k,0,n-1}]//Flatten

{T(n,k) = sum(j=0,k, (-1)^j*binomial(n+1, j)*(k-j+2)^n)};
for(n=1,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 23 2019

[[sum((-1)^j*binomial(n+1, j)*(k-j+2)^n for j in (0..k)) for k in (0..n-1)] for n in (1..12)] # G. C. Greubel, Feb 23 2019

T(n,k) = Sum_{j=0..k} (-1)^j *binomial(n+1,j)*(k+2-j)^n, for k=0..n-1.

A306547 Triangle read by rows, defined by Riordan's general Eulerian recursion: T(n, k) = (k+3)T(n-1, k) + (n-k-2) T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-2)^(n-1).

Original entry on oeis.org

1, 1, -2, 1, -11, 4, 1, -55, 35, -8, 1, -274, 210, -91, 16, 1, -1368, 986, -637, 219, -32, 1, -6837, 3180, -3473, 1752, -507, 64, 1, -34181, -1431, -17951, 10543, -4563, 1147, -128, 1, -170900, -145310, -129950, 48442, -30524, 11470, -2555, 256, 1, -854494, -1726360, -1490890, -2314, -177832, 84176, -28105, 5627, -512

Offset: 1

G. C. Greubel, Feb 22 2019

Row sums are {1, -1, -6, -27, -138, -831, -5820, -46563, -419070, -4190703, ...}.

The Mathematica code for e(n,k,m) gives eleven sequences of which the first few are in the OEIS (see Crossrefs section).

			Triangle begins with:
1.
1,      -2.
1,     -11,       4.
1,     -55,      35,      -8.
1,    -274,     210,     -91,    16.
1,   -1368,     986,    -637,   219,    -32.
1,   -6837,    3180,   -3473,  1752,   -507,    64.
1,  -34181,   -1431,  -17951, 10543,  -4563,  1147,  -128.
1, -170900, -145310, -129950, 48442, -30524, 11470, -2555, 256.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 214-215.

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A157011 (m=0), A008292 (m=1), A157012 (m=2), A157013 (m=3), this sequence.

e[n_, 0, m_]:= 1; (* Example for m=3 *)
e[n_, k_, m_]:= 0 /; k >= n;
e[n_, k_, m_]:= (k+m)*e[n-1, k, m] + (n-k+1-m)*e[n-1, k-1, m];
Table[Flatten[Table[Table[e[n, k, m], {k,0,n-1}], {n,1,10}]], {m,0,10}]
T[n_, 1]:= 1; T[n_, n_]:= (-2)^(n-1); T[n_, k_]:= T[n, k] = (k+3)*T[n-1, k] + (n-k-2)*T[n-1, k-1]; Table[T[n, k], {n, 1, 12}, {k, 1, n}]//Flatten

{T(n, k) = if(k==1, 1, if(k==n, (-2)^(n-1), (k+3)*T(n-1, k) + (n-k-2)* T(n-1, k-1)))};
for(n=1, 12, for(k=1, n, print1(T(n, k), ", ")))

def T(n, k):
    if (k==1): return 1
    elif (k==n): return (-2)^(n-1)
    else: return (k+3)*T(n-1, k) + (n-k-2)* T(n-1, k-1)
[[T(n, k) for k in (1..n)] for n in (1..12)]

T(n, k) = (k+3)*T(n-1, k) + (n-k-2)*T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-2)^(n-1).

e(n,k,m)= (k+m)*e(n-1, k, m) + (n-k+1-m)*e(n-1, k-1, m) with m=3.

cp's OEIS Frontend

A157012 Riordan's general Eulerian recursion: T(n,k) = (k+2)*T(n-1, k) + (n-k) * T(n-1, k-1), with T(n,0) = 1, T(n,n) = 0.

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A157013 Riordan's general Eulerian recursion: T(n, k) = (k+2)*T(n-1, k) + (n-k-1) * T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-1)^(n-1).

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A180246 Triangle T(n,k) read by rows: T(n,k) = Sum_{j=0..k} (-1)^j *binomial(n+1,j)*(k+2-j)^n, 0 <= k < n.

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A306547 Triangle read by rows, defined by Riordan's general Eulerian recursion: T(n, k) = (k+3)*T(n-1, k) + (n-k-2) * T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-2)^(n-1).

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A157012 Riordan's general Eulerian recursion: T(n,k) = (k+2)T(n-1, k) + (n-k) T(n-1, k-1), with T(n,0) = 1, T(n,n) = 0.

A157013 Riordan's general Eulerian recursion: T(n, k) = (k+2)T(n-1, k) + (n-k-1) T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-1)^(n-1).

A180246 Triangle T(n,k) read by rows: T(n,k) = Sum_{j=0..k} (-1)^j binomial(n+1,j)(k+2-j)^n, 0 <= k < n.

A306547 Triangle read by rows, defined by Riordan's general Eulerian recursion: T(n, k) = (k+3)T(n-1, k) + (n-k-2) T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-2)^(n-1).