Real-rooted atlas

A038314 → x ↦ x+1 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(34 + 36\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1156 + 2448\,x + 1296\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01156244812964900
13930412484813219246656343000
21336336565977689890566345216167961624010000
345435424240540480509379840539343360285534720604661761680700000
415448044169814051584259783718403667534848029124541440123350999042176782336117649000000
5525233501443892907128321236570499584218218323456023105469542401467876888576518074195968783641640968235430000000
61785793904896151267248414725605786264780811871076796006415711719288832013308750456422470458090651648213150526341122821109907456576480100000000
760716992766464578597225186304245052942431846460542491659632649615572204765184101811940991631367186725246468096326120305301913686325963168153610155995666841640353607000000000

All 15 rows

Technical details
Node id
n-abd52b1e4c5ddae9 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-a7a3fcc1bd498552

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:              1156,               2448,                1296;
  1:             39304,             124848,              132192,                46656;
  2:           1336336,            5659776,             8989056,              6345216,              1679616;
  3:          45435424,          240540480,           509379840,            539343360,            285534720,             60466176;
  4:        1544804416,         9814051584,         25978371840,          36675348480,          29124541440,          12335099904,           2176782336;
  5:       52523350144,       389290712832,       1236570499584,        2182183234560,        2310546954240,        1467876888576,         518074195968,          78364164096;
  6:     1785793904896,     15126724841472,      56057862647808,      118710767960064,      157117192888320,      133087504564224,       70458090651648,       21315052634112,        2821109907456;
  7:    60716992766464,    578597225186304,    2450529424318464,     6054249165963264,     9615572204765184,    10181194099163136,     7186725246468096,     3261203053019136,      863259631681536,     101559956668416;
  8:  2064377754059776,  21858117395927040,  104147500533534720,   294063530918215680,   544882424936693760,   692321198743093248,   610871645949788160,   369603012675502080,   146754137385861120,   34530385267261440,    3656158440062976;
  9: 70188843638032384, 817493590607671296, 4327907244393553920, 13747470070426583040, 29112289560903352320, 43154688054986145792, 45693199117044154368, 34557881685159444480, 18295349127437352960, 6457182044977889280, 1367403256583553024, 131621703842267136;
%F Formula
The row polynomials satisfy P(n,x) = (34+36*x)*P(n-1,x) for n >= 1, with P(0,x) = 1156 + 2448*x + 1296*x^2.
%t Mathematica
P[0, x_] := 1156 + 2448*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[(34 + 36*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] = 1156 + 2448*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = (34 + 36*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

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. [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.