A156919 -id:A156919 - cp's OEIS frontend

A156924 Fifth right hand column (n-m=4) of the A156920 triangle.

1, 83, 2685, 56285, 919615, 12813843, 160206627, 1854550395, 20291056470, 212826091180, 2161547322134, 21414479565774, 208076662576370, 1991164206775450, 18825064380813450

Offset: 0

Johannes W. Meijer, Feb 20 2009

Other columns A000340, A156922, A156923.
Equals A156920 fifth right hand column.
Equals A156919 fifth right hand column divided by 16.
Equals A142963 fifth right hand column divided by 2^n

a(n)=55*a(n-1)-1365*a(n-2)+20251*a(n-3)-200557*a(n-4)+1402203*a(n-5)-7137473*a(n-6)+26886431*a(n-7)-75433971*a(n-8)+157376597*a(n-9)-241846607*a(n-10)+268663713*a(n-11)-208880991*a(n-12)+107416665*a(n-13)-32730075*a(n-14)+4465125*a(n-15)
a(n)= (16*n^4-7776*n^3*3^n+256*n^3-104976*n^2*3^n+225000*n^2*5^n+ 1496*n^2- 464616*n*3^n+ 2250000*n*5^n-2016840*n*7^n+3776*n-673596*3^n+5568750*5^n-11092620*7^n+6200145*9^n+3465)/6144
G.f.: GF1(z;RHCnr=5) = (1+28*z-515*z^2+1654*z^3+8689*z^4-65864*z^5+142371*z^6-82242*z^7-99090*z^8+113400*z^9)/((1-9*z)*(1-7*z)^2*(1-5*z)^3*(1-3*z)^4*(1-z)^5)

A142966 Fourth column (m=3) of triangle A142963 divided by 8.

Original entry on oeis.org

1, 58, 877, 8030, 56285, 335162, 1792749, 8904486, 41949645, 190129090, 837258109, 3607669966, 15289404989, 63975698570, 265065915725, 1089837752118, 4454225465325, 18119738464530, 73441531708765, 296814738679390, 1196884383319261, 4817845684107098

Offset: 0

Wolfdieter Lang, Sep 15 2008

Cf. A142963.
Column m=2: 4*A142965.
From Johannes W. Meijer, Feb 20 2009: (Start)
Cf. A156925.
Equals A156920(n+3,3).
Equals A156919(n+3,3)/2^n.
(End)

a(n) = A142963(n+4,3)/8.
From Johannes W. Meijer, Feb 20 2009: (Start)
a(n) = 20a(n-1) - 175*a(n-2) + 882*a(n-3) - 2835*a(n-4) + 6072*a(n-5) - 8777*a(n-6) + 8458*a(n-7) - 5204*a(n-8) + 1848*a(n-9) - 288*a(n-10).
a(n) = -(105/2) - (143/3)*n - 14*n^2 - (4/3)*n^3 + 756*2^n + 48*2^n*n^2 + 384*2^n*n - (3645/2)*3^n - 405*3^n*n + 1120*4^n.
G.f.: (1 + 38*z - 108*z^2 - 242*z^3 + 839*z^4 - 444*z^5 - 180*z^6)/((1-z)^4*(1-2*z)^3*(1-3*z)^2*(1-4*z)).
(End)

A162007 Third left hand column of the EG1 triangle A162005.

Original entry on oeis.org

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.

A185411 A triangular decomposition of the double factorial numbers A001147.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 10, 1, 0, 8, 60, 36, 1, 0, 16, 296, 516, 116, 1, 0, 32, 1328, 5168, 3508, 358, 1, 0, 64, 5664, 42960, 64240, 21120, 1086, 1, 0, 128, 23488, 320064, 900560, 660880, 118632, 3272, 1, 0, 256, 95872, 2225728, 10725184, 14713840, 6049744, 638968, 9832, 1

Offset: 0

Paul Barry, Jan 26 2011

Row sums are A001147. Reversal of A185410. Contains A156919 as submatrix.
Row n counts perfect matchings of [2n] by number of matches in which the smaller entry is odd. For example, T(2,1)=2 counts 13/24, 14/23, in each of which only the first matching pair has an odd smaller entry. Outline proof. Consider the map on perfect matchings of [2n] given by "delete the entries n and n-1 and, if they were not originally matched to each other, match up their now-unmatched partners". Consideration of this map and its effect on the statistic "number of matches in which the smaller entry is odd" yields the Mathematica recurrence below. - David Callan, Dec 13 2011
Triangle T(n,k), 0 <= k <= n, given by (0, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, ...) DELTA (1, 0, 3, 0, 5, 0, 7, 0, 9, 11, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 12 2013
T(n,k), 0 <= k <= n, is the number of signed permutations of [n] that are products of balanced cycles (i.e., cuspidal elements of the type B Coxeter group) and have excedance number of type B equal to k. - Jose Bastidas, Jul 05 2023

			Triangle T(n,k) begins:
  1;
  0,   1;
  0,   2,     1;
  0,   4,    10,       1;
  0,   8,    60,      36,        1;
  0,  16,   296,     516,      116,        1;
  0,  32,  1328,    5168,     3508,      358,       1;
  0,  64,  5664,   42960,    64240,    21120,    1086,      1;
  0, 128, 23488,  320064,   900560,   660880,  118632,   3272,    1;
  0, 256, 95872, 2225728, 10725184, 14713840, 6049744, 638968, 9832, 1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Jose Bastidas, Christophe Hohlweg, and Franco Saliola, The Primitive Eulerian polynomial, arXiv:2306.15556 [math.CO], 2023. See Table 2 p. 18.
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, arXiv: 1204.4963v3 [math.CO], 2012.
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, Elect. J. Combinat. 20 (1) (2013) #P11.
Shi-Mei Ma, T. Mansour, and D. Callan, Some combinatorial arrays related to the Lotka-Volterra system, arXiv:1404.0731 [math.CO], 2014.
Shi-Mei Ma, T. Mansour and D. G. L. Wang, Combinatorics of Dumont differential system on the Jacobi elliptic functions, arXiv:1403.0233 [math.CO], 2014.
Shi-Mei Ma, Toufik Mansour, David G.L. Wang, and Yeong-Nan Yeh, Several variants of the Dumont differential system and permutation statistics, Science China Mathematics 60 (2018).
Shi-Mei Ma and Y.-N. Yeh, Stirling permutations, cycle structures of permutations and perfect matchings, arXiv:1503.06601 [math.CO], 2015.

Columns 0-1 give: A000007, A131577.

Cf. A001147, A185410, A156919 (another version).

u[n_, 0] := If[n==0, 1, 0]; u[n_, m_] /; m==1 := 2^(n - 1); u[n_, m_] /; m==n>=1 := 1; u[n_, m_] /; 1David Callan, Dec 13 2011 *)

G.f.: 1/(1-xy/(1-2x/(1-3xy/(1-4x/(1-5xy/(1-6x/(1-7xy/(1- ... (continued fraction).
T(n,k) = (2n-2k+1)*T(n-1,k-1) + 2k*T(n-1,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or k > n. - Philippe Deléham, Feb 12 2013
T(n,k) = 2^(n-k)*A211399(n,k). - Philippe Deléham, Feb 12 2013

Sequence terms corrected by Paul Barry, Jan 27 2011

A142968 Fifth column (m=4) of triangle A142963 divided by 16=2^4.

Original entry on oeis.org

1, 179, 5280, 82610, 919615, 8284857, 64730022, 457217400, 2999230965, 18608607535, 110625457964, 636103699038, 3562753619915, 19541111960965, 105392471360850, 560747327119908, 2950726075955265, 15387821226034875, 79656442803398680, 409857988825489610

Offset: 0

Wolfdieter Lang, Sep 15 2008

Column m=3: 8*A142966.
From Johannes W. Meijer, Feb 20 2009: (Start)
Cf. A156925.
Equals A156920(n+4,4).
Equals A156919(n+4,4)/2^n.
(End)

a(n) = A142963(n+5,3)/2^4.
From Johannes W. Meijer, Feb 20 2009: (Start)
a(n) = 35a(n-1) - 560a(n-2) + 5432a(n-3) - 35714a(n-4) + 168542a(n-5) - 589632a(n-6) + 1556776a(n-7) - 3126949a(n-8) + 4777591a(n-9) - 5506936a(n-10) + 4703032a(n-11) - 2881136a(n-12) + 1195632a(n-13) - 300672a(n-14) + 34560a(n-15).
a(n) = (1155/8) + (472/3)*n - 5544*2^n + (120285/4)*3^n - 49280*4^n + (196875/8)*5^n - 64*2^n*n^3 - 864*2^n*n^2 - 3824*2^n*n + (187/3)*n^2 + 1215*3^n*n^2 + 12150*3^n*n - 8960*4^n*n + (32/3)*n^3 + (2/3)*n^4.
G.f.: (1 + 144*z - 425*z^2 - 7382*z^3 + 48451*z^4 - 96764*z^5 - 2559*z^6 + 257002*z^7 - 312444*z^8 + 88344*z^9 + 30240*z^10)/((1-z)^5*(1-2*z)^4*(1-3*z)^3*(1-4*z)^2*(1-5*z)).
(End)

A186695 A Galton triangle: T(n,k) = (2k-1)*(T(n-1,k) + T(n-1,k-1)): a type B analog of the ordered Bell numbers A019538.

Original entry on oeis.org

1, 1, 3, 1, 12, 15, 1, 39, 135, 105, 1, 120, 870, 1680, 945, 1, 363, 4950, 17850, 23625, 10395, 1, 1092, 26565, 159600, 373275, 374220, 135135, 1, 3279, 138285, 1303155, 4795875, 8222445, 6621615, 2027025

Offset: 1

Peter Bala, Mar 26 2011

The row polynomials R(n,x) of A019538 satisfy the recurrence relation R(n+1,x) = x*d/dx((1+x)*R(n,x)), and have the expansion R(n,x) = Sum_{k = 1..n} k!*Stirling2(n,k)*x^k.
Here we consider a sequence of polynomials P(n,x) (n>=1) defined by means of the similar recursion P(n+1,x) = x*d/dx((1+x^2)*P(n,x)), with starting value P(1,x) = x.
The first few polynomials are P(1,x) = x, P(2,x) = x + 3*x^3, P(3,x) = x + 12*x^3 + 15*x^5, and P(4,x) = x + 39*x^3 + 135*x^5 + 105*x^7.
Clearly, the P(n,x) are odd polynomials of the form P(n,x) = Sum_{k = 1..n} T(n,k)*x^(2*k-1).
This triangle lists the coefficients T(n,k). They are related to A039755, the type B Stirling numbers of Suter, by T(n,k) = (2*k-1)!!*A039755(n-1,k-1).

			Triangle begins
  n\k.|..1.....2.....3......4......5......6
  =========================================
  ..1.|..1
  ..2.|..1.....3
  ..3.|..1....12....15
  ..4.|..1....39...135....105
  ..5.|..1...120...870...1680....945
  ..6.|..1...363..4950..17850..23625..10395
  ..
Examples of recurrence relation
  T(4,3) = 5*(T(3,3)+T(3,2)) = 5*(15+12) = 135;
  T(6,4) = 7*(T(5,4)+T(5,3)) = 7*(1680+870) = 17850.

Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 8.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Tech Report TR 99-05, 1999.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 1-3, 33-51 (2001).

Cf. A001147, A019538, A039755, A124212, A145901, A156919, A187075.

A186695 := proc(n, k) option remember; if k < 1 or k > n then 0; elif k = 1 then 1; else (2*k-1)*(procname(n-1, k) + procname(n-1, k-1)) ; end if; end proc: seq(seq(A186695(n,k),k = 1..n),n = 1..10);

T[n_, k_] := (2k-1)! Sum[(-1)^(k-j-1) (2j+1)^(n-1) Binomial[k-1, j], {j, 0, k-1}] / (2^(k-1) (k-1)!)^2;
Table[T[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 02 2019 *)

T(n+1,k+1) = ((2*k+1)!/(2^k*k!)^2)*Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*(2*j+1)^n.
Recurrence relation: T(n,k) = (2k-1)*(T(n-1,k) + T(n-1,k-1)) with boundary conditions T(n,1) = 1, T(1,k) = 0 for k >= 2.
E.g.f.: F(x,t) = (x/(1+x))*(exp(t)/sqrt((1+x) - x*exp(2*t)) - 1) = Sum_{n>=1} R(n,x)*t^n/n! = x*t + (x + 3*x^2)*t^2/2! + (x + 12*x^2 + 15*x^3)*t^3/3! + ....
Compare with the e.g.f. for A019538, which is (x/(1+x))*(exp(t)/((1+x) - x*exp(t))-1).
The row polynomials R(n,x) are related to the polynomials P(n,x) of the comments section by P(n,x) = 1/x*R(n,x^2).
The generating function F(x,t) satisfies the partial differential equation d/dt(F) = 2*x*(1+x)*d/dx(F) + (x-1)*F + x.
It follows that the polynomials P(n,x) := Sum_{k = 1..n} T(n,k)*x^(2*k-1) satisfy the recurrence P(n+1,x) = x*d/dx((1+x^2)*P(n,x)), with P(1,x) = x. (Cf. the recurrence relation for row polynomials of A185896.)
The recurrence relation for T(n,k) given above now follows.
The row polynomials R(n,x) = Sum_{k = 1..n} T(n,k)*x^k satisfy R(n,-x-1) = (-1)^n*((1+x)/x)*S(n,x), where S(n,x) is the n-th row polynomial of A187075.
In addition, R(n,1/(x-1)) = (1/(x-1)^n)*Q(n-1,x), where Q(n,x) is the n-th row polynomial of A156919.
Row sums are [1,4,28,280,3616...] = 1/2*A124212(n) for n >= 1.
Main diagonal is [1,3,15,105,...] = A001147(k) for k >= 1.
Put S(n) = sum {k = 1..n} (-1)^k*T(n,k)/(k+1). Then for m>=2, S(2*m-1) = S(2*m) = (4^m-1)*Bernoulli(2*m)/m.
From Peter Bala, Aug 30 2016: (Start)
n-th row polynomial R(n,x) = 1/(1 + x)^(3/2) * Sum_{k >= 0} (1/4)^k*(x/(1 + x))^k*binomial(2*k,k)*(2*k + 1)^n.
R(n,x) = (1/(1 + x))*Sum_{k = 0..n} binomial(2*k,k)*A145901(n,k)*(x/4)^k. (End)

A187075 A Galton triangle: T(n,k) = 2kT(n-1,k) + (2k-1)T(n-1,k-1).

Original entry on oeis.org

1, 2, 3, 4, 18, 15, 8, 84, 180, 105, 16, 360, 1500, 2100, 945, 32, 1488, 10800, 27300, 28350, 10395, 64, 6048, 72240, 294000, 529200, 436590, 135135, 128, 24384, 463680, 2857680, 7938000, 11060280, 7567560, 2027025, 256, 97920, 2904000, 26107200, 105099120, 220041360, 249729480, 145945800, 34459425

Offset: 1

Peter Bala, Mar 27 2011

This is a companion triangle to A186695.
Let f(x) = (exp(2*x) + 1)^(-1/2); then the n-th derivative of f equals Sum_{k=1..n} (-1)^k*T(n,k)*(f(x))^(2*k+1). - Groux Roland, May 17 2011
Triangle T(n,k), 1 <= k <= n, given by (0, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, ...) DELTA (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 20 2013

			Triangle begins
n\k.|...1.....2......3......4......5......6
===========================================
..1.|...1
..2.|...2.....3
..3.|...4....18.....15
..4.|...8....84....180....105
..5.|..16...360...1500...2100....945
..6.|..32..1488..10800..27300..28350..10395
..
Examples of recurrence relation:
T(4,3) = 6*T(3,3) + 5*T(3,2) = 6*15 + 5*18 = 180;
T(6,4) = 8*T(5,4) + 7*T(5,3) = 8*2100 + 7*1500 = 27300.

G. C. Greubel, Table of n, a(n) for the first 50 rows
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, arXiv:1204.4963v3 [math.CO]
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, El J. Combinat. 20 (1) (2013) P11.
E. Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 (2001) 33-51.

Cf. A075497, A156919, A186695, A211402.

A187075 := proc(n, k) option remember; if k < 1 or k > n then 0; elif k = 1 then 2^(n-1); else 2*k*procname(n-1, k) + (2*k-1)*procname(n-1, k-1) ; end if; end proc:seq(seq(A187075(n,k),k = 1..n),n = 1..10);

Flatten[Table[2^(n - 2*k)*Binomial[2 k, k]*k!*StirlingS2[n, k], {n, 10}, {k, 1, n}]] (* G. C. Greubel, Jun 17 2016 *)

# uses[delehamdelta from A084938]
# Adds a first column (1,0,0,0, ...).
def A187075_triangle(n):
    return delehamdelta([(i+1)*int(is_even(i+1)) for i in (0..n)], [i+1 for i in (0..n)])
A187075_triangle(4)  # Peter Luschny, Oct 20 2013

T(n,k) = 2^(n-2*k)*binomial(2k,k)*k!*Stirling2(n,k).
Recurrence relation T(n,k) = 2*k*T(n-1,k) + (2*k-1)*T(n-1,k-1) with boundary conditions T(1,1) = 1, T(1,k) = 0 for k >= 2.
G.f.: F(x,t) = 1/sqrt((1+x)-x*exp(2*t)) - 1 = Sum_{n >= 1} R(n,x)*t^n/n! = x*t + (2*x+3*x^2)*t^2/2! + (4*x+18*x^2+15*x^3)*t^3/3! + ....
The g.f. F(x,t) satisfies the partial differential equation dF/dt = 2*(x+x^2)*dF/dx + x*F.
The row polynomials R(n,x) satisfy the recursion R(n+1,x) = 2*(x+x^2)*R'(n,x) + x*R(n,x) where ' indicates differentiation with respect to x.
O.g.f. for column k: (2k-1)!!*x^k/Product_{m = 1..k} (1-2*m*x) (compare with A075497). T(n,k) = (2*k-1)!!*A075497(n,k).
The row polynomials R(n,x) = Sum_{k = 1..n} T(n,k)*x^k satisfy R(n,-x-1) = (-1)^n*(1+x)/x*P(n,x) where P(n,x) is the n-th row polynomial of A186695. We also have R(n,x/(1-x)) = (x/(1-x)^n)*Q(n-1,x) where Q(n,x) is the n-th row polynomial of A156919.
T(n,k) = 2^(n-k)*A211608(n,k). - Philippe Deléham, Oct 20 2013

A185410 A decomposition of the double factorials A001147.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 10, 4, 0, 1, 36, 60, 8, 0, 1, 116, 516, 296, 16, 0, 1, 358, 3508, 5168, 1328, 32, 0, 1, 1086, 21120, 64240, 42960, 5664, 64, 0, 1, 3272, 118632, 660880, 900560, 320064, 23488, 128, 0, 1, 9832, 638968, 6049744, 14713840, 10725184, 2225728, 95872, 256, 0

Offset: 0

Paul Barry, Jan 26 2011

Row sums are A001147. Reversal of A185411.
From Peter Bala, Jul 24 2012: (Start)
This is the case k = 2 of the 1/k—Eulerian polynomials introduced by Savage and Viswanathan. They give a combinatorial interpretation of the triangle in terms of an ascent statistic on sets of inversion sequences and a geometric interpretation in terms of lecture hall polytopes.
Row reverse of A156919.
(End)
Triangle T(n,k), 0<=k<=n, given by (1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...) DELTA (0, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 12 2013

			Triangle begins:
  1,
  1,    0,
  1,    2,      0,
  1,   10,      4,       0,
  1,   36,     60,       8,        0,
  1,  116,    516,     296,       16,        0,
  1,  358,   3508,    5168,     1328,       32,       0,
  1, 1086,  21120,   64240,    42960,     5664,      64,     0,
  1, 3272, 118632,  660880,   900560,   320064,   23488,   128,   0,
  1, 9832, 638968, 6049744, 14713840, 10725184, 2225728, 95872, 256, 0,
  ...
In the Savage-Viswanathan paper, the coefficients appear as
  1
  1    2
  1   10     4
  1   36    60     8
  1  116   516   296    16
  1  358  3508  5168  1328   32
  1 1086 21120 64240 42960 5664 64
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
S.-M. Ma and T. Mansour, The 1/k-Eulerian polynomials and k-Stirling permutations, arXiv preprint arXiv:1409.6525 [math.CO], 2014.
C. D. Savage and G. Viswanathan, The 1/k-Eulerian polynomials, Elec. J. of Comb., Vol. 19, Issue 1, #P9 (2012).

Cf. A156919, A001147 (row sums), A112857, A173018, A186695, A202038 (alt. row sums).

T[0, 0] := 1;  T[n_, -1] := 0;  T[n_, n_] := 0; T[n_, k_] := T[n, k] = (n - k)*T[n - 1, k - 1] + (2*k + 1)*T[n - 1, k]; Join[{1}, Table[If[k < 0, 0, If[k >= n, 0, 2^k*T[n, k]]], {n, 1, 5}, {k, 0, n}] // Flatten] (* G. C. Greubel, Jun 30 2017 *)

G.f.: 1/(1-x/(1-2xy/(1-3x/(1-4xy/(1-5x/(1-6xy/(1-7x/(1-8xy/(1- .... (continued fraction).
From Peter Bala, Jul 24 2012: (Start)
T(n,k) = sum {j=0..k}(-1)^(k-j)/4^j*C(n+1/2,k-j)*C(2*j,j)*(2*j+1)^n.
Recurrence equation: T(n+1,k) = (2*k+1)*T(n,k) + 2*(n-k+1)*T(n,k-1).
E.g.f.: sqrt(E(x,2*z)) = 1 + z + (1+2*x)*z^2/2! + (1+10*x+4*x^2)*z^3/3! + ..., where E(x,z) = (1-x)/(exp(z*(x-1)) - x) is the e.g.f. for the Eulerian numbers (version A173018). Cf. A156919.
Row polynomial R(n,x) = sum {k = 1..n} 2^(n-2*k)*C(2*k,k)*k!*Stirling2(n,k)*(x-1)^(n-k). R(n,4*x)/(1-4*x)^(n+1/2) = sum {k>=0} C(2*k,k)*(2*k+1)^n*x^k. The sequence of rational functions x*R(n,x)/(1-x)^(n+1) conjecturally occurs in the first column of (I - x*A112857)^(-1). (1+x)^(n-1)*R(n,x/(x+1)) gives the n-th row polynomial of A186695.
Row sums A001147. Alt. row sums A202038. (End)
T(n,k) = 2^k*A102365(n,k). - Philippe Deléham, Feb 12 2013

A142964 a(n) = 62^n - 2n - 5.

Original entry on oeis.org

1, 5, 15, 37, 83, 177, 367, 749, 1515, 3049, 6119, 12261, 24547, 49121, 98271, 196573, 393179, 786393, 1572823, 3145685, 6291411, 12582865, 25165775, 50331597, 100663243, 201326537, 402653127, 805306309, 1610612675, 3221225409, 6442450879, 12884901821

Offset: 0

Wolfdieter Lang, Sep 15 2008

Previous name was: One half of second column (m=1) of triangle A142963.

Essentially a duplicate of A050488. - Johannes W. Meijer, Feb 20 2009

			a(3) = 6*2^3 - 2*3 - 5 = 37.

Eric Billault, Walter Damin, Robert Ferréol, and Rodolphe Garin, MPSI Classes Prépas - Khôlles de Maths, Exercices corrigés, Ellipses, 2012, exercice 2.22 (1) pp 26, 43-44.

Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A142965 (m=2 column/4).
Equals A050488(n+1).
Equals A156920(n+1,1).
Equals A156919(n+1,1)/2^n.
Cf. A156925, A339771.
Partial sums of A033484.

seq(6*2^n-2*n-5,n=0..40); # Bernard Schott, Dec 16 2020

a[n_]:=6*2^n-2n-5;Array[a,32,0] (* or *) CoefficientList[Series[(1+x)/((1-x)^2*(1-2*x)),{x,0,31}],x] (* or *) LinearRecurrence[{4,-5,2},{1,5,15},32] (* James C. McMahon, Aug 12 2025 *)

Vec((1+z)/((1-z)^2*(1-2*z)) + O(z^50)) \\ Michel Marcus, Jun 18 2017

a(n) = A142693(n+2,1)/2.
From Johannes W. Meijer, Feb 20 2009: (Start)
a(n) = 4a(n-1) - 5a(n-2) + 2a(n-3) for n > 2 with a(0) = 1, a(1) = 5, a(2) = 15.
G.f.: (1+z)/((1-z)^2*(1-2*z)). (End)
a(n) = Sum_{i=0..n} Sum_{j=0..n} 2^min(i,j) (Billault et al) (compare with A339771 that has max instead of min). - Bernard Schott, Dec 16 2020
a(n) = 2*A066524(n+1) - A339771(n). - Kevin Ryde, Dec 17 2020
E.g.f.: 6*exp(2*x) - exp(x)*(5 + 2*x). - Stefano Spezia, Dec 17 2020

New name using a formula of Bernard Schott by Peter Luschny, Dec 17 2020

cp's OEIS Frontend

A156924 Fifth right hand column (n-m=4) of the A156920 triangle.

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A142966 Fourth column (m=3) of triangle A142963 divided by 8.

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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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A185411 A triangular decomposition of the double factorial numbers A001147.

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A142968 Fifth column (m=4) of triangle A142963 divided by 16=2^4.

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A186695 A Galton triangle: T(n,k) = (2k-1)*(T(n-1,k) + T(n-1,k-1)): a type B analog of the ordered Bell numbers A019538.

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A187075 A Galton triangle: T(n,k) = 2*k*T(n-1,k) + (2*k-1)*T(n-1,k-1).

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Sage

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A185410 A decomposition of the double factorials A001147.

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A187075 A Galton triangle: T(n,k) = 2kT(n-1,k) + (2k-1)T(n-1,k-1).

A142964 a(n) = 62^n - 2n - 5.