A172431 → ⊙ C(n,k+1) (shift -1) → D+1

nonnegative coefficients

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

Recurrence

\(\left(60\,n - 144\,n x + 70\,n x^{2} - 8\,n x^{3} - 28\,n^{2} - 28\,n^{2} x + 46\,n^{2} x^{2} - 8\,n^{2} x^{3} - 164\,n^{3} + 380\,n^{3} x - 164\,n^{3} x^{2} + 16\,n^{3} x^{3} + 212\,n^{4} - 256\,n^{4} x + 48\,n^{4} x^{2} - 96\,n^{5} + 48\,n^{5} x + 16\,n^{6}\right)\,P_{n}(x) = \left(816\,x - 1464\,x^{2} + 744\,x^{3} - 96\,x^{4} + 24\,n - 4132\,n x + 7644\,n x^{2} - 3706\,n x^{3} + 440\,n x^{4} - 16\,n^{2} + 7808\,n^{2} x - 12522\,n^{2} x^{2} + 4908\,n^{2} x^{3} - 464\,n^{2} x^{4} - 60\,n^{3} - 7556\,n^{3} x + 9240\,n^{3} x^{2} - 2492\,n^{3} x^{3} + 144\,n^{3} x^{4} + 100\,n^{4} + 4112\,n^{4} x - 3240\,n^{4} x^{2} + 432\,n^{4} x^{3} - 64\,n^{5} - 1192\,n^{5} x + 432\,n^{5} x^{2} + 16\,n^{6} + 144\,n^{6} x\right)\,P_{n-1}(x) + \left(-120\,x + 1014\,x^{2} - 1746\,x^{3} + 996\,x^{4} - 144\,x^{5} + 500\,n x - 4182\,n x^{2} + 7016\,n x^{3} - 3698\,n x^{4} + 472\,n x^{5} - 776\,n^{2} x + 6132\,n^{2} x^{2} - 8220\,n^{2} x^{3} + 2852\,n^{2} x^{4} - 208\,n^{2} x^{5} + 588\,n^{3} x - 4312\,n^{3} x^{2} + 3856\,n^{3} x^{3} - 624\,n^{3} x^{4} - 224\,n^{4} x + 1520\,n^{4} x^{2} - 624\,n^{4} x^{3} + 32\,n^{5} x - 208\,n^{5} x^{2}\right)\,P_{n-1}'(x) + \left(-30\,x^{2} + 188\,x^{3} - 318\,x^{4} + 192\,x^{5} - 32\,x^{6} + 108\,n x^{2} - 662\,n x^{3} + 1059\,n x^{4} - 576\,n x^{5} + 80\,n x^{6} - 140\,n^{2} x^{2} + 770\,n^{2} x^{3} - 900\,n^{2} x^{4} + 240\,n^{2} x^{5} + 82\,n^{3} x^{2} - 388\,n^{3} x^{3} + 240\,n^{3} x^{4} - 20\,n^{4} x^{2} + 80\,n^{4} x^{3}\right)\,P_{n-1}''(x)\)

\(P_{1}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	1														3
3	5	6	1													12
4	10	29	13	1												53
5	17	100	102	24	1											244
6	26	265	540	280	40	1										1152
7	37	586	2075	2112	658	62	1									5531
8	50	1141	6321	11165	6720	1386	91	1								26875
9	65	2024	16268	45136	46970	18480	2688	128	1							131760
10	82	3345	36912	149352	244608	165594	45540	4884	174	1						650492
11	101	5230	76005	424320	1017240	1084356	510510	102960	8415	230	1					3229368
12	122	7821	144925	1070289	3554496	5560632	4120116	1415700	217074	13871	297	1				16105344
13	145	11276	259666	2454760	10825155	23535072	25635456	13846976	3602170	431860	22022	376	1			80624935
14	170	15769	441948	5209776	29509480	85463235	129698712	103093848	42095196	8532810	817960	33852	468	1		404913225
15	197	21490	720447	10366720	73443370	274019746	555449895	616346016	370528158	117689572	19025006	1485120	50596	574	1	2039146908

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Technical details

Node id: n-a05d8afbacbf6624 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-ae44fb5a0d72a25c

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
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 ?

A262442: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:  1;
   2:  2,    1;
   3:  5,    6,     1;
   4: 10,   29,    13,      1;
   5: 17,  100,   102,     24,      1;
   6: 26,  265,   540,    280,     40,      1;
   7: 37,  586,  2075,   2112,    658,     62,     1;
   8: 50, 1141,  6321,  11165,   6720,   1386,    91,    1;
   9: 65, 2024, 16268,  45136,  46970,  18480,  2688,  128,   1;
  10: 82, 3345, 36912, 149352, 244608, 165594, 45540, 4884, 174, 1;

%F Formula

The row polynomials satisfy (16*n^6-96*n^5+212*n^4-164*n^3-28*n^2+60*n+(48*n^5-256*n^4+380*n^3-28*n^2-144*n)*x+(48*n^4-164*n^3+46*n^2+70*n)*x^2+(16*n^3-8*n^2-8*n)*x^3)*P(n,x) = (16*n^6-64*n^5+100*n^4-60*n^3-16*n^2+24*n+(144*n^6-1192*n^5+4112*n^4-7556*n^3+7808*n^2-4132*n+816)*x+(432*n^5-3240*n^4+9240*n^3-12522*n^2+7644*n-1464)*x^2+(432*n^4-2492*n^3+4908*n^2-3706*n+744)*x^3+(144*n^3-464*n^2+440*n-96)*x^4)*P(n-1,x) + ((32*n^5-224*n^4+588*n^3-776*n^2+500*n-120)*x-(208*n^5-1520*n^4+4312*n^3-6132*n^2+4182*n-1014)*x^2-(624*n^4-3856*n^3+8220*n^2-7016*n+1746)*x^3-(624*n^3-2852*n^2+3698*n-996)*x^4-(208*n^2-472*n+144)*x^5)*d/dx P(n-1,x) - ((20*n^4-82*n^3+140*n^2-108*n+30)*x^2-(80*n^4-388*n^3+770*n^2-662*n+188)*x^3-(240*n^3-900*n^2+1059*n-318)*x^4-(240*n^2-576*n+192)*x^5-(80*n-32)*x^6)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^6 - 64*n^5 + 100*n^4 - 60*n^3 - 16*n^2 + 24*n + (144*n^6 - 1192*n^5 + 4112*n^4 - 7556*n^3 + 7808*n^2 - 4132*n + 816)*x + (432*n^5 - 3240*n^4 + 9240*n^3 - 12522*n^2 + 7644*n - 1464)*x^2 + (432*n^4 - 2492*n^3 + 4908*n^2 - 3706*n + 744)*x^3 + (144*n^3 - 464*n^2 + 440*n - 96)*x^4)*P[n - 1, x] + ((32*n^5 - 224*n^4 + 588*n^3 - 776*n^2 + 500*n - 120)*x - (208*n^5 - 1520*n^4 + 4312*n^3 - 6132*n^2 + 4182*n - 1014)*x^2 - (624*n^4 - 3856*n^3 + 8220*n^2 - 7016*n + 1746)*x^3 - (624*n^3 - 2852*n^2 + 3698*n - 996)*x^4 - (208*n^2 - 472*n + 144)*x^5)*D[P[n - 1, x], x] - ((20*n^4 - 82*n^3 + 140*n^2 - 108*n + 30)*x^2 - (80*n^4 - 388*n^3 + 770*n^2 - 662*n + 188)*x^3 - (240*n^3 - 900*n^2 + 1059*n - 318)*x^4 - (240*n^2 - 576*n + 192)*x^5 - (80*n - 32)*x^6)*D[P[n - 1, x], {x, 2}])/(16*n^6 - 96*n^5 + 212*n^4 - 164*n^3 - 28*n^2 + 60*n + (48*n^5 - 256*n^4 + 380*n^3 - 28*n^2 - 144*n)*x + (48*n^4 - 164*n^3 + 46*n^2 + 70*n)*x^2 + (16*n^3 - 8*n^2 - 8*n)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((16*n^6 - 64*n^5 + 100*n^4 - 60*n^3 - 16*n^2 + 24*n + (144*n^6 - 1192*n^5 + 4112*n^4 - 7556*n^3 + 7808*n^2 - 4132*n + 816)*x + (432*n^5 - 3240*n^4 + 9240*n^3 - 12522*n^2 + 7644*n - 1464)*x^2 + (432*n^4 - 2492*n^3 + 4908*n^2 - 3706*n + 744)*x^3 + (144*n^3 - 464*n^2 + 440*n - 96)*x^4)*v[i - 1] + ((32*n^5 - 224*n^4 + 588*n^3 - 776*n^2 + 500*n - 120)*x - (208*n^5 - 1520*n^4 + 4312*n^3 - 6132*n^2 + 4182*n - 1014)*x^2 - (624*n^4 - 3856*n^3 + 8220*n^2 - 7016*n + 1746)*x^3 - (624*n^3 - 2852*n^2 + 3698*n - 996)*x^4 - (208*n^2 - 472*n + 144)*x^5)*deriv(v[i - 1], x) - ((20*n^4 - 82*n^3 + 140*n^2 - 108*n + 30)*x^2 - (80*n^4 - 388*n^3 + 770*n^2 - 662*n + 188)*x^3 - (240*n^3 - 900*n^2 + 1059*n - 318)*x^4 - (240*n^2 - 576*n + 192)*x^5 - (80*n - 32)*x^6)*deriv(deriv(v[i - 1], x), x))/(16*n^6 - 96*n^5 + 212*n^4 - 164*n^3 - 28*n^2 + 60*n + (48*n^5 - 256*n^4 + 380*n^3 - 28*n^2 - 144*n)*x + (48*n^4 - 164*n^3 + 46*n^2 + 70*n)*x^2 + (16*n^3 - 8*n^2 - 8*n)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-f5f4cf0c3e186e1f
Coefficients of one sign. Every nonzero row, n >= 1: 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-f5f4cf0c3e186e1f
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-f5f4cf0c3e186e1f

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

preserver_image derivative_polynomial, shift 0 e-f5f4cf0c3e186e1f

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - preserver_image hadamard_multiplier, shift -1 e-1491609a6fc5f8c9
    
    Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
    
    premise (sequence): Even row Pascal-square read by antidiagonals.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check three_term_recurrence e-83282836b5ac538e root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

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.
[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