A083061 -id:A083061 - cp's OEIS frontend

A162007 Third left hand column of the EG1 triangle A162005.

1, 270, 36096, 4766048, 704357760, 120536980224, 24060789342208, 5590122715250688, 1503080384197754880, 464520829174515630080, 163839204411117787938816, 65500849343294249018327040

Offset: 3

Johannes W. Meijer, Jun 27 2009

Third left hand column of the EG1 triangle A162005.
Other left hand columns are A000182 and A162006.
Related to A094665, A083061 and A156919.
A000079, A036289 and A100381 appear in the a(n, 3) formula.
A001789, A003472, A054849, A002409, A054851, A140325 and A140354 (scaled by 2^(m-1)) appear one by one in the a(n, m) formulas for m= 4 and higher .

nmax := 14; mmax := nmax: imax := nmax: T1(0, x) := 1: T1(0, x+1) := 1: for i from 1 to imax do T1(i, x) := expand((2*x+1)*(x+1)*T1(i-1, x+1) - 2*x^2*T1(i-1, x)): dx := degree(T1(i, x)): for k from 0 to dx do c(k) := coeff(T1(i, x), x, k) od: T1(i, x+1) := sum(c(j1)*(x+1)^(j1), j1 = 0..dx): od: for i from 0 to imax do for j from 0 to i do A083061(i, j) := coeff(T1(i, x), x, j) od: od: for n from 0 to nmax do for k from 0 to n do A094665(n+1, k+1) := A083061(n, k) od: od: A094665(0, 0) := 1: for n from 1 to nmax do A094665(n, 0) := 0 od: for m from 1 to mmax do A156919(0, m) := 0 end do: for n from 0 to nmax do A156919(n, 0) := 2^n end do: for n from 1 to nmax do for m from 1 to mmax do A156919(n, m) := (2*m+2)*A156919(n-1, m) + (2*n-2*m+1) * A156919(n-1, m-1) end do end do: m:=3; for n from m to nmax do a(n, m) := sum((-1)^(m-p1-1)*sum(2^(n-q-1)*binomial(n-q-1, m-p1-1) * A094665(n-1, q) * A156919(q, p1), q=1..n-m+p1), p1=0..m-1) od: seq(a(n, m), n = m..nmax);
# Maple program edited by Johannes W. Meijer, Sep 25 2012

a(n) = sum((-1)^(m-p-1)*sum(2^(n-q-1)*binomial(n-q-1,m-p-1)*A094665(n-1,q)* A156919(q,p),q=1..n-m+p), p=0..m-1) with m = 3.

A162006 Second left hand column of the EG1 triangle A162005.

Original entry on oeis.org

1, 28, 1032, 52736, 3646208, 330545664, 38188155904, 5488365862912, 961530104709120, 201865242068910080, 50052995352723193856, 14476381898608390176768, 4831399425299156001882112

Offset: 2

Johannes W. Meijer, Jun 27 2009

Second left hand column of the EG1 triangle A162005.
Other left hand columns are A000182 and A162007.
Related to A094665, A083061 and A156919.
A000079 and A036289 appear in the Maple program.

nmax := 14; mmax := nmax: imax := nmax: T1(0, x) := 1: T1(0, x+1) := 1: for i from 1 to imax do T1(i, x) := expand((2*x+1)*(x+1)*T1(i-1, x+1) - 2*x^2*T1(i-1, x)): dx := degree(T1(i, x)): for k from 0 to dx do c(k) := coeff(T1(i, x), x, k) od: T1(i, x+1) := sum(c(j1)*(x+1)^(j1), j1 = 0..dx): od: for i from 0 to imax do for j from 0 to i do A083061(i, j) := coeff(T1(i, x), x, j) od: od: for n from 0 to nmax do for k from 0 to n do A094665(n+1, k+1) := A083061(n, k) od: od: A094665(0, 0) := 1: for n from 1 to nmax do A094665(n, 0) := 0 od: for m from 1 to mmax do A156919(0, m) := 0 end do: for n from 0 to nmax do A156919(n, 0) := 2^n end do: for n from 1 to nmax do for m from 1 to mmax do A156919(n, m) := (2*m+2)*A156919(n-1, m) + (2*n-2*m+1) * A156919(n-1, m-1) end do end do: m:=2; for n from m to nmax do a(n, m) := sum((-1)^(m-p1-1)*sum(2^(n-q-1)*binomial(n-q-1, m-p1-1) * A094665(n-1, q) * A156919(q, p1), q=1..n-m+p1), p1=0..m-1) od: seq(a(n, m), n = m..nmax);
# Maple program edited by Johannes W. Meijer, Sep 25 2012

a(n) = sum((-1)^(m-p-1)*sum(2^(n-q-1)*binomial(n-q-1,m-p-1)*A094665(n-1,q)*A156919(q,p),q=1..n-m+p), p=0..m-1) with m = 2.

A085734 Triangle read by rows: T(0,0) = 1, T(n,k) = Sum_{j=max(0,1-k)..n-k} (2^j)(binomial(k+j,1+j) + binomial(k+j+1,1+j))T(n-1,k-1+j).

Original entry on oeis.org

1, 2, 3, 16, 30, 15, 272, 588, 420, 105, 7936, 18960, 16380, 6300, 945, 353792, 911328, 893640, 429660, 103950, 10395, 22368256, 61152000, 65825760, 36636600, 11351340, 1891890, 135135, 1903757312, 5464904448, 6327135360, 3918554640, 1427025600, 310269960, 37837800, 2027025

Offset: 0

Philippe Deléham, Jul 20 2003

A triangle related to Euler numbers and tangent numbers.
T(n,k) = number of down-up permutations on [2n+2] with k+1 left-to-right maxima. For example, T(1,1) counts the following 3 down-up permutations on [4] each with 2 left-to-right maxima: 2143, 3142, 3241. - David Callan, Oct 25 2004
It appears that Sum_{k=0..n} (-1)^(n-k)*T(n,k)*x^(k+1) is the zeta polynomial for the poset of even-sized subsets of [2n+2] ordered by inclusion. - Geoffrey Critzer, Apr 22 2023

			Triangle begins as:
    1;
    2,   3;
   16,  30,  15;
  272, 588, 420, 105; ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles , JIS 9 (2006) 06.4.1.
Tian Han, Sergey Kitaev, and Philip B. Zhang, Distribution of maxima and minima statistics on alternating permutations, Springer numbers, and avoidance of flat POPs, arXiv:2408.12865 [math.CO], 2024. See p. 4.
Alan D. Sokal, The Euler and Springer numbers as moment sequences, arXiv:1804.04498 [math.CO], 2018.
M. S. Tokmachev, Correlations Between Elements and Sequences in a Numerical Prism, Bulletin of the South Ural State University, Ser. Mathematics. Mechanics. Physics, 2019, Vol. 11, No. 1, 24-33.

T(n, 0) = A000182(n), tangent numbers, T(n, n) = A001147(n+1), Sum_{k>=0} T(n, k) = A000364(n+1), Euler numbers.
Cf. A088874.
A subtriangle of A098906.

t[n_, k_]:= t[n, k] = Sum[(2^j)*(Binomial[k+j, 1+j] + Binomial[k+j+1, 1+j])*t[n-1, k-1+j], {j, Max[0, 1-k], n-k}]; t[0, 0] = 1; Table[t[n, k], {n,0,7}, {k,0,n}]//Flatten (* Jean-François Alcover, Feb 26 2013 *)

T(n,m):=sum((stirling1(k,m)*sum((i-k)^(2*n)*binomial(2*k,i)*(-1)^(n+m+i),i,0,k-1))/(2^(k-1)*k!),k,1,n); /* Vladimir Kruchinin, May 20 2013 */

{T(n,k) = if(n==0 && k==0, 1, sum(j=max(0, 1-k), n-k, (2^j)*(binomial(k+j,1+j) + binomial(k+j+1,1+j))*T(n-1,k-1+j)))};
for(n=0,5, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Mar 21 2019

@CachedFunction
def T(n,k):
    if n==0 and k==0: return 1
    else: return sum((2^j)*(binomial(k+j,1+j) + binomial(k+j+1,1+j))*T(n-1,k-1+j) for j in (max(0, 1-k)..(n-k)))
[[T(n, k) for k in (0..n)] for n in (0..7)] # G. C. Greubel, Mar 21 2019

T(n, k) = A083061(n, k)*2^(n-k). - Philippe Deléham, Feb 27 2005
E.g.f.: sec(x)^y. - Vladeta Jovovic, May 20 2007
T(n,m) = Sum_{k=1..n} (Stirling1(k,m)*Sum_{i=0..k-1} (i-k)^(2*n)* binomial(2*k,i)*(-1)^(n+m+i))/(2^(k-1)*k!). - Vladimir Kruchinin, May 20 2013

Edited and extended by Ray Chandler, Nov 23 2003

A094346 Another version of triangular array in A036970: triangle T(n,k), 0<=k<=n, read by rows; given by [0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 3, 8, 6, 0, 17, 54, 60, 24, 0, 155, 556, 762, 480, 120, 0, 2073, 8146, 12840, 10248, 4200, 720, 0, 38227, 161424, 282078, 263040, 139440, 40320, 5040, 0, 929569, 4163438, 7886580, 8240952, 5170800, 1965600, 423360, 40320

Offset: 0

Philippe Deléham, Jun 08 2004, Jun 13 2007

Diagonals: A000007, A001469, A005440; A000182, A005990. Row sums: A001469.

			Triangle begins:
1;
0, 1;
0, 1, 2;
0, 3, 8, 6;
0, 17, 54, 60, 24;
0, 155, 556, 762, 480, 120;
0, 2073, 8146, 12840, 10248, 4200, 720;
0, 38227, 161424, 282078, 263040, 139440, 40320, 5040;
0, 929569, 4163438, 7886580, 8240952, 5170800, 1965600, 423360, 40320; ...

D. Dumont, Sur une conjecture de Gandhi concernant les nombres de Genocchi, Discrete Mathematics 1 (1972) 321-327.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
J. M. Gandhi, Research Problems: A Conjectured Representation of Genocchi Numbers, Amer. Math. Monthly 77 (1970), no. 5, 505-506. MR1535914
Andrei K. Svinin, Tuenter polynomials and a Catalan triangle, arXiv:1603.05748 [math.CO], 2016. (Has a signed version of this triangle, see p. 1).

Cf. A094665, A083061.

G[_, 1] = 1;
G[x_, n_] := G[x, n] = (x+1)^2 G[x+1, n-1] - x^2 G[x, n-1] // Expand;
row[0] = {1};
row[n_] := CoefficientList[x G[x, n], x];
Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Aug 17 2018 *)

{T(n, k) = local( A = x); if( k<0 || k>n, 0, for( j = 1, n, A = x^2 * ( subst(A, x, x+1) - A)); polcoeff( A, k+1))} /* Michael Somos, Apr 10 2011 */

For n>=1, Sum_{k =1..n} T(n, k)*x^(k-1) = G(x, n), n-th Gandhi polynomial; the Gandhi polynomials are defined by G(x, n) = (x+1)^2*G(x+1, n-1) - x^2*G(x, n-1), G(x, 1) = 1. Sum_{k =0..n} T(n, k)*2^(2n-k) = A000182(n+1), tangent numbers. Sum_{k =0..n} T(n, k) = A001469(n+1), Genocchi numbers of first kind.

Sum_{k = 0..n} T(n, k)*2^(n-k) = A002105(n+1). - Philippe Deléham, Jun 10 2004

cp's OEIS Frontend

A162007 Third left hand column of the EG1 triangle A162005.

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A162006 Second left hand column of the EG1 triangle A162005.

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A085734 Triangle read by rows: T(0,0) = 1, T(n,k) = Sum_{j=max(0,1-k)..n-k} (2^j)*(binomial(k+j,1+j) + binomial(k+j+1,1+j))*T(n-1,k-1+j).

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A094346 Another version of triangular array in A036970: triangle T(n,k), 0<=k<=n, read by rows; given by [0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, ...] where DELTA is the operator defined in A084938.

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Programs

Mathematica

PARI

Formula

A085734 Triangle read by rows: T(0,0) = 1, T(n,k) = Sum_{j=max(0,1-k)..n-k} (2^j)(binomial(k+j,1+j) + binomial(k+j+1,1+j))T(n-1,k-1+j).