Real-rooted atlas

A108426: Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have k peaks of the form Ud.

A108426 ↗ nonnegative coefficients

Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard kernel

Recurrence from A108426

\(\left(2\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(6 + 5\,x + x^{2} - 27\,n - 23\,n x - 2\,n x^{2} + 27\,n^{2} + 24\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}(x) + \left(3\,x + 4\,x^{2} + x^{3} - 9\,n x - 10\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
235210
3122821566
4551651808414498
527310011430990330424066
61428618810920100105005128713234970
7775238760813969282061880240245005429312066
8432632451575969048139606783003527161113841944814302862562
924667515622754326300686439668643964476780189924050388075582486226824386
1014307151001500531081050562419006561555051482970274575849806280223839029393016796255680170
118414640645122402217608254506752256009003005511706203543239701593520504903140098062801144066587862471150402
125006710841722590015735376803548173200532225980055883727904206302100228342114088580992523902807542493880445740020801224161357010
133008305722707475148111148979762753690940045894849000542870499604683588624029804654880139709319754757127375114171057018253053017383860742900238552980386
1418227665201762007636078215948720211183061544387168946164509432823900495664369200361913666400199052516520819628009202483721240053625799507784390256786391526744402375085745978

b-files: A027307 (row sums), A104978 (reversed triangle), A108426 ?

Technical details
Node id
n-8e6b51ac707373f6 · revision 1 · JSON
Definition source
oeis_catalog:A108426
Fingerprint
f6-da93e8b7035ccffa

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

  • A104978 reversed triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
  • A108426 triangle 46 terms, 9 rows · tabl · nonn · catalog b-file?

Row sums ? A027307

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      1,       1;
  2:      3,       5,       2;
  3:     12,      28,      21,       5;
  4:     55,     165,     180,      84,      14;
  5:    273,    1001,    1430,     990,     330,      42;
  6:   1428,    6188,   10920,   10010,    5005,    1287,     132;
  7:   7752,   38760,   81396,   92820,   61880,   24024,    5005,    429;
  8:  43263,  245157,  596904,  813960,  678300,  352716,  111384,  19448,  1430;
  9: 246675, 1562275, 4326300, 6864396, 6864396, 4476780, 1899240, 503880, 75582, 4862;
%F Formula
The row polynomials satisfy (4*n^2+2*n)*P(n,x) = (27*n^2-27*n+6+(24*n^2-23*n+5)*x+(n^2-2*n+1)*x^2)*P(n-1,x) - ((9*n-3)*x+(10*n-4)*x^2+(n-1)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((27*n^2 - 27*n + 6 + (24*n^2 - 23*n + 5)*x + (n^2 - 2*n + 1)*x^2)*P[n - 1, x] - ((9*n - 3)*x + (10*n - 4)*x^2 + (n - 1)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 2*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((27*n^2 - 27*n + 6 + (24*n^2 - 23*n + 5)*x + (n^2 - 2*n + 1)*x^2)*v[i - 1] - ((9*n - 3)*x + (10*n - 4)*x^2 + (n - 1)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 2*n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A104978 reversed triangle, offset 0, from P0
%e Example
Triangle begins:
  0:    1;
  1:    1,     1;
  2:    2,     5,      3;
  3:    5,    21,     28,      12;
  4:   14,    84,    180,     165,      55;
  5:   42,   330,    990,    1430,    1001,     273;
  6:  132,  1287,   5005,   10010,   10920,    6188,    1428;
  7:  429,  5005,  24024,   61880,   92820,   81396,   38760,    7752;
  8: 1430, 19448, 111384,  352716,  678300,  813960,  596904,  245157,   43263;
  9: 4862, 75582, 503880, 1899240, 4476780, 6864396, 6864396, 4326300, 1562275, 246675;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (4*n^2+2*n)*P(n,x) = (27*n^2-27*n+6+(24*n^2-23*n+5)*x+(n^2-2*n+1)*x^2)*P(n-1,x) - ((9*n-3)*x+(10*n-4)*x^2+(n-1)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((27*n^2 - 27*n + 6 + (24*n^2 - 23*n + 5)*x + (n^2 - 2*n + 1)*x^2)*P[n - 1, x] - ((9*n - 3)*x + (10*n - 4)*x^2 + (n - 1)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 2*n)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((27*n^2 - 27*n + 6 + (24*n^2 - 23*n + 5)*x + (n^2 - 2*n + 1)*x^2)*v[i - 1] - ((9*n - 3)*x + (10*n - 4)*x^2 + (n - 1)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 2*n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A108426 triangle, offset 0, from P0
%e Example
Triangle begins:
  0:      1;
  1:      1,       1;
  2:      3,       5,       2;
  3:     12,      28,      21,       5;
  4:     55,     165,     180,      84,      14;
  5:    273,    1001,    1430,     990,     330,      42;
  6:   1428,    6188,   10920,   10010,    5005,    1287,     132;
  7:   7752,   38760,   81396,   92820,   61880,   24024,    5005,    429;
  8:  43263,  245157,  596904,  813960,  678300,  352716,  111384,  19448,  1430;
  9: 246675, 1562275, 4326300, 6864396, 6864396, 4476780, 1899240, 503880, 75582, 4862;
%F Formula
The row polynomials satisfy (4*n^2+2*n)*P(n,x) = (27*n^2-27*n+6+(24*n^2-23*n+5)*x+(n^2-2*n+1)*x^2)*P(n-1,x) - ((9*n-3)*x+(10*n-4)*x^2+(n-1)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((27*n^2 - 27*n + 6 + (24*n^2 - 23*n + 5)*x + (n^2 - 2*n + 1)*x^2)*P[n - 1, x] - ((9*n - 3)*x + (10*n - 4)*x^2 + (n - 1)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 2*n)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((27*n^2 - 27*n + 6 + (24*n^2 - 23*n + 5)*x + (n^2 - 2*n + 1)*x^2)*v[i - 1] - ((9*n - 3)*x + (10*n - 4)*x^2 + (n - 1)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 2*n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Used in 6

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.