A133214 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(30 + 28\,x + 16\,x^{2} + 63\,n + 46\,n x + 8\,n x^{2} + 40\,n^{2} + 16\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(214 + 476\,x + 432\,x^{2} + 128\,x^{3} + 461\,n + 990\,n x + 576\,n x^{2} + 96\,n x^{3} + 296\,n^{2} + 592\,n^{2} x + 192\,n^{2} x^{2} + 56\,n^{3} + 96\,n^{3} x\right)\,P_{n-1}(x) + \left(-37 - 252\,x - 484\,x^{2} - 288\,x^{3} - 64\,x^{4} - 36\,n - 232\,n x - 384\,n x^{2} - 128\,n x^{3} - 8\,n^{2} - 48\,n^{2} x - 64\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 45 + 64\,x + 16\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	45	64	16															125
1	291	708	432	64														1495
2	1857	6528	7104	2560	256													18305
3	11703	54180	87360	57600	14080	1024												225947
4	73021	420672	907920	903680	410880	73728	4096											2793997
5	452035	3120292	8474928	11439680	7965440	2688000	372736	16384										34529495
6	2780673	22388224	73459456	125970432	120243200	62849024	16515072	1835008	65536									426106625
7	17017191	156637764	603217152	1258521600	1538546688	1112444928	456916992	96731136	8847360	262144								5249142955
8	103697613	1074450240	4753576080	11704166400	17516835840	16335470592	9353379840	3118202880	545587200	41943040	1048576							64548358301
9	629632611	7253748612	36266733360	103034226240	182875376640	209936584704	155994144768	72979906560	20231946240	2984509440	196083712	4194304						792387087191
10	3811308225	48334108032	269559880128	868524771840	1785752305920	2441880207360	2244601970688	1370201260032	536141168640	125954949120	15917383680	905969664	16777216					9711602060545
11	23009818263	318557865444	1960928513856	7068414642432	16535658796800	26287383462912	28897776500736	21919422480384	11245612105728	3747722035200	757529051136	83097550848	4143972352	67108864				118849323904955
12	138596794461	2080102968384	14010164527632	55879269788160	146642027820288	266033064321024	341114870992896	310560257998848	198775606149120	87236397760512	25124863475712	4424453849088	425956737024	18790481920	268435456			1452464692100525
13	833135832963	13474433404260	98576741657520	431106053851200	1254684866722560	2559800693818368	3755893248307200	4000529678745600	3083623863091200	1694520475648000	645219971235840	162533411389440	25198623457280	2149161369600	84557168640	1073741824		17728229989441495
14	4999217131521	86680436078592	684517382691840	3257549594664960	10415923177820160	23617716826669056	39061397582118912	47730661706956800	43163020748390400	28651991911628800	13706249623306240	4580912796794880	1019486991810560	140402480906240	10694468567040	377957122048	4294967296	216132587197625345

Technical details

Node id: n-6956b1e0dc94fc10 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-56becc4b901ca19b

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:        45,         64,          16;
  1:       291,        708,         432,           64;
  2:      1857,       6528,        7104,         2560,          256;
  3:     11703,      54180,       87360,        57600,        14080,         1024;
  4:     73021,     420672,      907920,       903680,       410880,        73728,         4096;
  5:    452035,    3120292,     8474928,     11439680,      7965440,      2688000,       372736,       16384;
  6:   2780673,   22388224,    73459456,    125970432,    120243200,     62849024,     16515072,     1835008,       65536;
  7:  17017191,  156637764,   603217152,   1258521600,   1538546688,   1112444928,    456916992,    96731136,     8847360,     262144;
  8: 103697613, 1074450240,  4753576080,  11704166400,  17516835840,  16335470592,   9353379840,  3118202880,   545587200,   41943040,   1048576;
  9: 629632611, 7253748612, 36266733360, 103034226240, 182875376640, 209936584704, 155994144768, 72979906560, 20231946240, 2984509440, 196083712, 4194304;

%F Formula

The row polynomials satisfy (8*n^3+40*n^2+63*n+30+(16*n^2+46*n+28)*x+(8*n+16)*x^2)*P(n,x) = (56*n^3+296*n^2+461*n+214+(96*n^3+592*n^2+990*n+476)*x+(192*n^2+576*n+432)*x^2+(96*n+128)*x^3)*P(n-1,x) - (8*n^2+36*n+37+(48*n^2+232*n+252)*x+(64*n^2+384*n+484)*x^2+(128*n+288)*x^3+64*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 45 + 64*x + 16*x^2.

%t Mathematica

P[0, x_] := 45 + 64*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((56*n^3 + 296*n^2 + 461*n + 214 + (96*n^3 + 592*n^2 + 990*n + 476)*x + (192*n^2 + 576*n + 432)*x^2 + (96*n + 128)*x^3)*P[n - 1, x] - (8*n^2 + 36*n + 37 + (48*n^2 + 232*n + 252)*x + (64*n^2 + 384*n + 484)*x^2 + (128*n + 288)*x^3 + 64*x^4)*D[P[n - 1, x], x])/(8*n^3 + 40*n^2 + 63*n + 30 + (16*n^2 + 46*n + 28)*x + (8*n + 16)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 45 + 64*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((56*n^3 + 296*n^2 + 461*n + 214 + (96*n^3 + 592*n^2 + 990*n + 476)*x + (192*n^2 + 576*n + 432)*x^2 + (96*n + 128)*x^3)*v[i - 1] - (8*n^2 + 36*n + 37 + (48*n^2 + 232*n + 252)*x + (64*n^2 + 384*n + 484)*x^2 + (128*n + 288)*x^3 + 64*x^4)*deriv(v[i - 1], x))/(8*n^3 + 40*n^2 + 63*n + 30 + (16*n^2 + 46*n + 28)*x + (8*n + 16)*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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013609 of Delannoy paths counted by number of weak peaks.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Delannoy paths counted by number of weak peaks.?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Delannoy paths counted by number of weak peaks.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

A104684 → basis A013611 → D+1

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023