Real-rooted atlas

A038334 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(44 + 72\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1936 + 6336\,x + 5184\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
019366336518413456
1851844181766842883732481560896
2374809624532992602173446569164826873856181063936
31649162241349314560441593856072260812805912248320193491763221003416576
47256313856712438087682914519449606358951526407804167782405108182548481393140695042436396322816
53192778096643657182183424179534398095364896392675328080122789232640786660112465924290873340723210030613004288282621973446656
61404822362521618390401836646410532684688261123447060443430912705080545247232092301453196001287551937079672832353077577750937672220413630873632784148919812096
7618121839509504910324890913996859584901950734336227505989266440192558423791835807744913784386640412672996855694516813824699093603946856448285992837978259456519986978142289923802961274698203136

All 15 rows

Technical details
Node id
n-1f571e6a0df94da8 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-5e3baf07c55bade9

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:                1936,                 6336,                  5184;
  1:               85184,               418176,                684288,                373248;
  2:             3748096,             24532992,              60217344,              65691648,               26873856;
  3:           164916224,           1349314560,            4415938560,            7226081280,             5912248320,             1934917632;
  4:          7256313856,          71243808768,          291451944960,          635895152640,           780416778240,           510818254848,            139314069504;
  5:        319277809664,        3657182183424,        17953439809536,        48963926753280,         80122789232640,         78666011246592,          42908733407232,          10030613004288;
  6:      14048223625216,      183904018366464,      1053268468826112,      3447060443430912,       7050805452472320,       9230145319600128,        7551937079672832,        3530775777509376,         722204136308736;
  7:     618121839509504,     9103248909139968,     59584901950734336,    227505989266440192,     558423791835807744,     913784386640412672,      996855694516813824,      699093603946856448,      285992837978259456,      51998697814228992;
  8:   27197360938418176,   445047724446842880,   3277169607290388480,  14300376468176240640,   40951078067959234560,   80413026024356315136,   109654126396849520640,   102533728578872279040,    62918424355217080320,   22879427038260756480,    3743906242624487424;
  9: 1196683881290399744, 21540309863227195392, 176238898880949780480, 865172776324662558720, 2831474540698895646720, 6486650765964742754304, 10614519435215033597952, 12406581158043545763840, 10150839129308355624960, 5536821343259103068160, 1812050621430251913216, 269561249468963094528;
%F Formula
The row polynomials satisfy P(n,x) = (44+72*x)*P(n-1,x) for n >= 1, with P(0,x) = 1936 + 6336*x + 5184*x^2.
%t Mathematica
P[0, x_] := 1936 + 6336*x + 5184*x^2;
P[n_, x_] := P[n, x] = Expand[(44 + 72*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] = 1936 + 6336*x + 5184*x^2; for(n = 1, N, my(i = n + 1); v[i] = (44 + 72*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.