Real-rooted atlas

A286718 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(1 + 2\,x + 3\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1426
22822454
3280276848648
4364041481644272169720
5582407364834600764080032174960
611065601515792804696214360304802208643674160
724344320355605442073489663253121099280109536582412888179840
8608608000937702240589493488199602592401326244936960364672148482562380855680
9170410240002747287872018381222144676785955215229186562185001282008473611450883686451271425670400
1052827174400088574128832062476364390424656609040060746197440981934128010596270727566720034329608960010242357047123200
11179612392960003117174729088023013446469376963277436140825585028937604553499984005566600300846919389442680550409912320214016204884853696435200
126645658539520001189277128354560913841013948672402439544310848113930155791936219649557283202970342108096284933746944193019143689028659202774323250380840963309294160972800
1326582634158080000489002168420864003893219481465600017925263800331264536208532029913611064585407166721627435957804801733803409395213419440686087471846553629154611207563878411714568192138990354760857600
14114305326879744000021558745925258752001771884810714380800848650733043556352266420196373525376583018878914151689210891699993984107102265760089692379663138048589678215526427480175923290833899522016501762695168163846254565964238592000
Technical details
Node id
n-b3407a472b6e181d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6c462b5a02443188

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:           4,           2;
  2:          28,          22,           4;
  3:         280,         276,          84,          8;
  4:        3640,        4148,        1644,        272,         16;
  5:       58240,       73648,       34600,       7640,        800,        32;
  6:     1106560,     1515792,      804696,     214360,      30480,      2208,       64;
  7:    24344320,    35560544,    20734896,    6325312,    1099280,    109536,     5824,     128;
  8:   608608000,   937702240,   589493488,  199602592,   40132624,   4936960,   364672,   14848,   256;
  9: 17041024000, 27472878720, 18381222144, 6767859552, 1522918656, 218500128, 20084736, 1145088, 36864, 512;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+1+2*x)*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[(3*n + 1 + 2*x)*P[n - 1, x]];
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] = (3*n + 1 + 2*x)*v[i - 1]); 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

Properties

Proof chains ? JSON

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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.