A038284 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(24 + 14\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 576 + 672\,x + 196\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	576	672	196															1444
1	13824	24192	14112	2744														54872
2	331776	774144	677376	263424	38416													2085136
3	7962624	23224320	27095040	15805440	4609920	537824												79235168
4	191102976	668860416	975421440	758661120	331914240	77446656	7529536											3010936384
5	4586471424	18728091648	32774160384	31863767040	18587197440	6505519104	1264962048	105413504										114415582592
6	110075314176	513684799488	1048773132288	1223568654336	892185477120	416353222656	121436356608	20239392768	1475789056									4347792138496
7	2641807540224	13869489586176	32362142367744	44048471556096	38542412611584	22483074023424	8743417675776	2185854418944	318770436096	20661046784								165216101262848
8	63403380965376	369853055631360	970864271032320	1510233310494720	1541696504463360	1079187553124352	524605060546560	174868353515520	38252452331520	4958651228160	289254654976							6278211847988224
9	1521681143169024	9764120668667904	28478685283614720	49837699246325760	58143982454046720	47484252337471488	27699147196858368	11541311332024320	3366215805173760	654541962117120	76363228913664	4049565169664						238572050223552512
10	36520347436056576	255642432052396032	820186136168103936	1594806375882424320	2093183368345681920	1953637810455969792	1329559065449201664	664779532724600832	242367537972510720	62836028363243520	10996304963567616	1166274768863232	56693912375296					9065737908494995456
11	876488338465357824	6646703233362296832	23263461316768038912	49757958927531638784	72563690102650306560	76191874607782821888	59260346917164417024	34568535701679243264	15123734369484668928	4901210212332994560	1143615716211032064	181938863942664192	17688500661092352	793714773254144				344498040522809827328
12	21035720123168587776	171791714339210133504	651376916869505089536	1519879472695511875584	2438139987449050300416	2844496652023892017152	2488934570520905515008	1659289713680603676672	846929124691141459968	329361326268777234432	96063720161726693376	20377152761578389504	2971668111063515136	266688163813392384	11112006825558016			13090925539866773438464
13	504857282956046106624	4417501225865403432960	18038130005617064017920	45596384180865356267520	79793672316514373468160	102401879472860112617472	99557382820836220600320	74668037115627165450240	43556354984115846512640	19761679576126634065920	6916587851644321923072	1833943748542055055360	356600173327621816320	48003869486410629120	4000322457200885760	155568095557812224		497455170514937390661632
14	12116574790945106558976	113088031382154327883776	494760137296925184491520	1346847040419407446671360	2553397514128459950981120	3574756519779843931373568	3823003500320110871052288	3185836250266759059210240	2090705039237560632606720	1084069279604661068759040	442661622505236603076608	140846879888029828251648	34233616639451694366720	6144495294260560527360	768061911782570065920	59738148694199894016	2177953337809371136	18903296479567620845142016

Technical details

Node id: n-76d494d4a790974a · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-48a0cbb267bcb941

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:              576,              672,               196;
  1:            13824,            24192,             14112,              2744;
  2:           331776,           774144,            677376,            263424,             38416;
  3:          7962624,         23224320,          27095040,          15805440,           4609920,            537824;
  4:        191102976,        668860416,         975421440,         758661120,         331914240,          77446656,           7529536;
  5:       4586471424,      18728091648,       32774160384,       31863767040,       18587197440,        6505519104,        1264962048,         105413504;
  6:     110075314176,     513684799488,     1048773132288,     1223568654336,      892185477120,      416353222656,      121436356608,       20239392768,       1475789056;
  7:    2641807540224,   13869489586176,    32362142367744,    44048471556096,    38542412611584,    22483074023424,     8743417675776,     2185854418944,     318770436096,     20661046784;
  8:   63403380965376,  369853055631360,   970864271032320,  1510233310494720,  1541696504463360,  1079187553124352,   524605060546560,   174868353515520,   38252452331520,   4958651228160,   289254654976;
  9: 1521681143169024, 9764120668667904, 28478685283614720, 49837699246325760, 58143982454046720, 47484252337471488, 27699147196858368, 11541311332024320, 3366215805173760, 654541962117120, 76363228913664, 4049565169664;

%F Formula

The row polynomials satisfy P(n,x) = (24+14*x)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 672*x + 196*x^2.

%t Mathematica

P[0, x_] := 576 + 672*x + 196*x^2;
P[n_, x_] := P[n, x] = Expand[(24 + 14*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] = 576 + 672*x + 196*x^2; for(n = 1, N, my(i = n + 1); v[i] = (24 + 14*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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

References

[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
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.