Real-rooted atlas

A243662: Triangle read by rows: the reversed x = 1+q Narayana triangle at m=2.

A243662 ↗ nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence from A243662

\(\left(n + 2\,n^{2}\right)\,P_{n}(x) = \left(3 + 2\,x - \frac{27}{2}\,n - 8\,n x + \frac{27}{2}\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(\frac{3}{2}\,x + x^{2} - \frac{9}{2}\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
3128121
45555151126
5273364156241818
61428238014003503515594
777521550411628408068048139693
843263100947921694189599751197631289510
92466756578007084003967041239702156019608012157150
101430715429214553281803552120138138031878042504303699116348960
1184146402804880039443625305370001425060041277607371007800045001201125642146
12500671081835793962884819082545428601388415604859454610972962156756613513564351431976789620
133008305721203322288208998081620711521601294470100532581984146198976265816323115035223300893216817668465964
14182276652078986549201502670936016529380296116553322605520946860179057736039790608059685912585156035464012090195160708178054
1511124755664519155264321073707478481298436950721020200461285474246377620528423916541408982499733233612578966410482472544544160162241484093913917

b-files: A102537 (reversed triangle), A243662 ?

Other recurrences

X ↦ x+1 substitution · proved equal for all n (proof step)

\(\left(2\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(6 + 4\,x - 27\,n - 16\,n x + 27\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x + 2\,x^{2} - 9\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-f00f6de2d4bbc6f5 · revision 2 · JSON
Definition source
oeis_catalog:A243662
derived:shift_x_plus_one (equal by e-50000157a371bcb3)
Fingerprint
f6-23ffd7df897ade57

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 -1)
  • 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 ?

  • A102537 reversed triangle 49 terms, 9 rows · tabl · nonn · catalog b-file?
  • A243662 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A003168

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       3,       1;
   3:      12,       8,       1;
   4:      55,      55,      15,       1;
   5:     273,     364,     156,      24,       1;
   6:    1428,    2380,    1400,     350,      35,      1;
   7:    7752,   15504,   11628,    4080,     680,     48,     1;
   8:   43263,  100947,   92169,   41895,    9975,   1197,    63,    1;
   9:  246675,  657800,  708400,  396704,  123970,  21560,  1960,   80,  1;
  10: 1430715, 4292145, 5328180, 3552120, 1381380, 318780, 42504, 3036, 99, 1;
%F Formula
The row polynomials satisfy (2*n^2+n)*P(n,x) = ((27/2)*n^2-(27/2)*n+3+(6*n^2-8*n+2)*x)*P(n-1,x) - (((9/2)*n-3/2)*x+(4*n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(((27/2)*n^2 - (27/2)*n + 3 + (6*n^2 - 8*n + 2)*x)*P[n - 1, x] - (((9/2)*n - 3/2)*x + (4*n - 1)*x^2)*D[P[n - 1, x], x])/(2*n^2 + n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (((27/2)*n^2 - (27/2)*n + 3 + (6*n^2 - 8*n + 2)*x)*v[i - 1] - (((9/2)*n - 3/2)*x + (4*n - 1)*x^2)*deriv(v[i - 1], x))/(2*n^2 + n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A102537 reversed triangle, offset 1, from P1
%e Example
Triangle begins:
   1: 1;
   2: 1,  3;
   3: 1,  8,   12;
   4: 1, 15,   55,    55;
   5: 1, 24,  156,   364,    273;
   6: 1, 35,  350,  1400,   2380,    1428;
   7: 1, 48,  680,  4080,  11628,   15504,    7752;
   8: 1, 63, 1197,  9975,  41895,   92169,  100947,   43263;
   9: 1, 80, 1960, 21560, 123970,  396704,  708400,  657800,  246675;
  10: 1, 99, 3036, 42504, 318780, 1381380, 3552120, 5328180, 4292145, 1430715;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (2*n^2+n)*P(n,x) = ((27/2)*n^2-(27/2)*n+3+(6*n^2-8*n+2)*x)*P(n-1,x) - (((9/2)*n-3/2)*x+(4*n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(((27/2)*n^2 - (27/2)*n + 3 + (6*n^2 - 8*n + 2)*x)*P[n - 1, x] - (((9/2)*n - 3/2)*x + (4*n - 1)*x^2)*D[P[n - 1, x], x])/(2*n^2 + n)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (((27/2)*n^2 - (27/2)*n + 3 + (6*n^2 - 8*n + 2)*x)*v[i - 1] - (((9/2)*n - 3/2)*x + (4*n - 1)*x^2)*deriv(v[i - 1], x))/(2*n^2 + n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A243662 triangle, offset 1, from P1
%e Example
Triangle begins:
   1:       1;
   2:       3,       1;
   3:      12,       8,       1;
   4:      55,      55,      15,       1;
   5:     273,     364,     156,      24,       1;
   6:    1428,    2380,    1400,     350,      35,      1;
   7:    7752,   15504,   11628,    4080,     680,     48,     1;
   8:   43263,  100947,   92169,   41895,    9975,   1197,    63,    1;
   9:  246675,  657800,  708400,  396704,  123970,  21560,  1960,   80,  1;
  10: 1430715, 4292145, 5328180, 3552120, 1381380, 318780, 42504, 3036, 99, 1;
%F Formula
The row polynomials satisfy (2*n^2+n)*P(n,x) = ((27/2)*n^2-(27/2)*n+3+(6*n^2-8*n+2)*x)*P(n-1,x) - (((9/2)*n-3/2)*x+(4*n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(((27/2)*n^2 - (27/2)*n + 3 + (6*n^2 - 8*n + 2)*x)*P[n - 1, x] - (((9/2)*n - 3/2)*x + (4*n - 1)*x^2)*D[P[n - 1, x], x])/(2*n^2 + n)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (((27/2)*n^2 - (27/2)*n + 3 + (6*n^2 - 8*n + 2)*x)*v[i - 1] - (((9/2)*n - 3/2)*x + (4*n - 1)*x^2)*deriv(v[i - 1], x))/(2*n^2 + n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Used in 7

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.