A133214 → basis A013610

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-1 - 12\,x - 36\,x^{2} - n - 12\,n x - 36\,n x^{2}\right)\,P_{n-2}(x) + \left(9 + 18\,x + 6\,n + 12\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 13 + 48\,x + 36\,x^{2}\)

\(P_{1}(x) = 63 + 342\,x + 540\,x^{2} + 216\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	13	48	36															97
1	63	342	540	216														1161
2	321	2304	5616	5184	1296													14721
3	1683	15030	49680	73440	45360	7776												192969
4	8989	96048	401220	820800	835920	373248	46656											2582881
5	48639	605094	3056508	8006040	11476080	8654688	2939328	279936										35066313
6	265729	3772416	22357440	71463168	132632640	142829568	83607552	22394880	1679616									481003009
7	1462563	23331942	158718528	599260032	1368093888	1926239616	1630347264	765904896	166281984	10077696								6649718409
8	8097453	143400240	1101206340	4797169920	13014146880	22699450368	25337007360	17400821760	6726862080	1209323520	60466176							92497952097
9	45046719	876878838	7502808060	37048671000	116509440960	242789145984	337854198528	308360701440	175981766400	57090147840	8646663168	362797056						1293068265993
10	251595969	5339568384	50371195248	278069267520	994950689040	2413626107904	4032350429184	4614657619968	3524346650880	1702727516160	470910578688	60949905408	2176782336					18150727906689
11	1409933619	32399826966	334077728400	2039005713888	8181335139120	22659131255904	44240171728896	61055049369600	58773531218688	38239867860480	15875273576448	3791954829312	424472555520	13060694016				255660741430857
12	7923848253	196009419312	2193108774948	14664880706112	65222870560848	203145718583808	454260715900992	735268041750528	857052209325312	706224860909568	397115299316736	143415127425024	29911166078976	2925595459584	78364164096			3611681892224097
13	44642381823	1182740176710	14271682447260	103762993141080	506707476977520	1753540398241632	4421236530110400	8220157500297600	11274870479558400	11292506069644800	8077583031598080	3972143429130240	1260824980746240	231762015313920	19982861844480	470184984576		51151047016594761
14	252055236609	7120742842368	92174933456640	723994588800000	3851338234947840	14663781855830016	41170813824172032	86520625287782400	136687964525798400	161588841283584000	140983497087713280	88552898569175040	38460609310556160	10824703200460800	1767895542005760	135413275557888	2821109907456	726034745427826689

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Technical details

Node id: n-0180c64ef6975bff · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-e47e283e05d357e9

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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:       13,        48,         36;
  1:       63,       342,        540,         216;
  2:      321,      2304,       5616,        5184,         1296;
  3:     1683,     15030,      49680,       73440,        45360,         7776;
  4:     8989,     96048,     401220,      820800,       835920,       373248,        46656;
  5:    48639,    605094,    3056508,     8006040,     11476080,      8654688,      2939328,       279936;
  6:   265729,   3772416,   22357440,    71463168,    132632640,    142829568,     83607552,     22394880,      1679616;
  7:  1462563,  23331942,  158718528,   599260032,   1368093888,   1926239616,   1630347264,    765904896,    166281984,    10077696;
  8:  8097453, 143400240, 1101206340,  4797169920,  13014146880,  22699450368,  25337007360,  17400821760,   6726862080,  1209323520,   60466176;
  9: 45046719, 876878838, 7502808060, 37048671000, 116509440960, 242789145984, 337854198528, 308360701440, 175981766400, 57090147840, 8646663168, 362797056;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = -(n+1+(12*n+12)*x+(36*n+36)*x^2)*P(n-2,x) + (6*n+9+(12*n+18)*x)*P(n-1,x) for n >= 2, with P(0,x) = 13 + 48*x + 36*x^2, P(1,x) = 63 + 342*x + 540*x^2 + 216*x^3.

%t Mathematica

P[0, x_] := 13 + 48*x + 36*x^2;
P[1, x_] := 63 + 342*x + 540*x^2 + 216*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (12*n + 12)*x + (36*n + 36)*x^2)*P[n - 2, x] + (6*n + 9 + (12*n + 18)*x)*P[n - 1, x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 13 + 48*x + 36*x^2; v[2] = 63 + 342*x + 540*x^2 + 216*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(n + 1 + (12*n + 12)*x + (36*n + 36)*x^2)*v[i - 2] + (6*n + 9 + (12*n + 18)*x)*v[i - 1])/(n + 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-f0f0fb352cf0ac03
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-f0f0fb352cf0ac03
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-f0f0fb352cf0ac03
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-f0f0fb352cf0ac03

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

basis_map A013610 e-f0f0fb352cf0ac03

Geometric basis map (elementary) polytool ?
- premise (sequence): Delannoy paths counted by number of weak peaks.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-4ab8ec5aa6b36e54 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023