Real-rooted atlas

A038284 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(49 + 210\,x + 144\,x^{2} + 126\,n x + 144\,n x^{2} + 36\,n^{2} x^{2}\right)\,P_{n}(x) = \left(686 + 5292\,x + 8568\,x^{2} + 3888\,x^{3} + 1764\,n x + 4536\,n x^{2} + 2592\,n x^{3} + 504\,n^{2} x^{2} + 432\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 196 + 1344\,x + 1296\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0196134412962836
12744282245443227648113048
238416526848152409615482885184004156048
353782492198403556224054190080362880008957952144755936
47529536154893312746807040151732224015240960007524679681463132164849429312
5105413504252992409614637417984371743948804978713600036870930432143386951682293235712157737147776
614757890564047878553627323180236883270644531213940398080001376514736128802966929408256842399744348285173765013085212928
720661046784637540872192491817244262417486835351552351298031616004336021418803233724611035136161810711838724388393189376515978035200156363280506368

All 15 rows

Technical details
Node id
n-f7ef7289ee552610 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-950975dd37953e66

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           196,            1344,             1296;
  1:          2744,           28224,            54432,            27648;
  2:         38416,          526848,          1524096,          1548288,            518400;
  3:        537824,         9219840,         35562240,         54190080,          36288000,           8957952;
  4:       7529536,       154893312,        746807040,       1517322240,        1524096000,         752467968,         146313216;
  5:     105413504,      2529924096,      14637417984,      37174394880,       49787136000,       36870930432,       14338695168,        2293235712;
  6:    1475789056,     40478785536,     273231802368,     832706445312,     1394039808000,     1376514736128,      802966929408,      256842399744,       34828517376;
  7:   20661046784,    637540872192,    4918172442624,   17486835351552,    35129803161600,    43360214188032,    33724611035136,    16181071183872,     4388393189376,     515978035200;
  8:  289254654976,   9917302456320,   86068017745920,  349736707031040,   819695407104000,  1214085997264896,  1180361386229760,   755116655247360,   307187523256320,   72236924928000,    7492001071104;
  9: 4049565169664, 152726457827328, 1472719414763520, 6732431610347520, 18033298956288000, 31161540596465664, 36355130695876608, 29071991227023360, 15768959527157760, 5562243219456000, 1153768164950016, 106993205379072;
%F Formula
The row polynomials satisfy (49+(126*n+210)*x+(36*n^2+144*n+144)*x^2)*P(n,x) = (686+(1764*n+5292)*x+(504*n^2+4536*n+8568)*x^2+(432*n^2+2592*n+3888)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 1344*x + 1296*x^2.
%t Mathematica
P[0, x_] := 196 + 1344*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((686 + (1764*n + 5292)*x + (504*n^2 + 4536*n + 8568)*x^2 + (432*n^2 + 2592*n + 3888)*x^3)*P[n - 1, x])/(49 + (126*n + 210)*x + (36*n^2 + 144*n + 144)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 196 + 1344*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((686 + (1764*n + 5292)*x + (504*n^2 + 4536*n + 8568)*x^2 + (432*n^2 + 2592*n + 3888)*x^3)*v[i - 1])/(49 + (126*n + 210)*x + (36*n^2 + 144*n + 144)*x^2)); 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.