A272866 → Veronese 2 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-18 + 84\,x - 98\,x^{2} + 46\,x^{3} - 4\,x^{4} + 43\,n - 137\,n x + 144\,n x^{2} - 54\,n x^{3} + 4\,n x^{4} - 34\,n^{2} + 66\,n^{2} x - 50\,n^{2} x^{2} + 8\,n^{2} x^{3} + 9\,n^{3} - 13\,n^{3} x + 4\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-2 + 88\,x + 258\,x^{2} - 344\,x^{3} + 142\,x^{4} - 12\,x^{5} + 9\,n + 77\,n x - 553\,n x^{2} + 506\,n x^{3} - 166\,n x^{4} + 12\,n x^{5} - 16\,n^{2} - 224\,n^{2} x + 350\,n^{2} x^{2} - 174\,n^{2} x^{3} + 24\,n^{2} x^{4} + 9\,n^{3} + 59\,n^{3} x - 55\,n^{3} x^{2} + 12\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-4\,x - 8\,x^{2} + 316\,x^{3} - 520\,x^{4} + 124\,x^{5} - 8\,x^{6} - 18\,n x + 190\,n x^{2} - 482\,n x^{3} + 176\,n x^{4} - 16\,n x^{5} + 18\,n^{2} x - 134\,n^{2} x^{2} + 74\,n^{2} x^{3} - 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 1 | 3 | ||||||
| 2 | 12 | 13 | 1 | 26 | |||||
| 3 | 31 | 90 | 33 | 1 | 155 | ||||
| 4 | 59 | 448 | 369 | 62 | 1 | 939 | |||
| 5 | 96 | 1465 | 2740 | 1065 | 100 | 1 | 5467 | ||
| 6 | 142 | 3681 | 13737 | 11069 | 2475 | 147 | 1 | 31252 | |
| 7 | 197 | 7798 | 50400 | 77665 | 34538 | 4977 | 203 | 1 | 175779 |
Compute more
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Technical details
- Node id
n-7986093a99b0748f· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-04b0847defb80657
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 0)
- 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 ExampleTriangle begins: 0: 1; 1: 2, 1; 2: 12, 13, 1; 3: 31, 90, 33, 1; 4: 59, 448, 369, 62, 1; 5: 96, 1465, 2740, 1065, 100, 1; 6: 142, 3681, 13737, 11069, 2475, 147, 1; 7: 197, 7798, 50400, 77665, 34538, 4977, 203, 1; 8: 261, 14680, 147994, 392988, 321155, 90188, 9030, 268, 1; 9: 334, 25353, 369972, 1544286, 2137698, 1071207, 206724, 15174, 342, 1;
%F FormulaThe row polynomials satisfy (9*n^3-34*n^2+43*n-18-(13*n^3-66*n^2+137*n-84)*x+(4*n^3-50*n^2+144*n-98)*x^2+(8*n^2-54*n+46)*x^3+(4*n-4)*x^4)*P(n,x) = (9*n^3-16*n^2+9*n-2+(59*n^3-224*n^2+77*n+88)*x-(55*n^3-350*n^2+553*n-258)*x^2+(12*n^3-174*n^2+506*n-344)*x^3+(24*n^2-166*n+142)*x^4+(12*n-12)*x^5)*P(n-1,x) + ((18*n^2-18*n-4)*x-(134*n^2-190*n+8)*x^2+(74*n^2-482*n+316)*x^3-(8*n^2-176*n+520)*x^4-(16*n-124)*x^5-8*x^6)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^3 - 16*n^2 + 9*n - 2 + (59*n^3 - 224*n^2 + 77*n + 88)*x - (55*n^3 - 350*n^2 + 553*n - 258)*x^2 + (12*n^3 - 174*n^2 + 506*n - 344)*x^3 + (24*n^2 - 166*n + 142)*x^4 + (12*n - 12)*x^5)*P[n - 1, x] + ((18*n^2 - 18*n - 4)*x - (134*n^2 - 190*n + 8)*x^2 + (74*n^2 - 482*n + 316)*x^3 - (8*n^2 - 176*n + 520)*x^4 - (16*n - 124)*x^5 - 8*x^6)*D[P[n - 1, x], x])/(9*n^3 - 34*n^2 + 43*n - 18 - (13*n^3 - 66*n^2 + 137*n - 84)*x + (4*n^3 - 50*n^2 + 144*n - 98)*x^2 + (8*n^2 - 54*n + 46)*x^3 + (4*n - 4)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((9*n^3 - 16*n^2 + 9*n - 2 + (59*n^3 - 224*n^2 + 77*n + 88)*x - (55*n^3 - 350*n^2 + 553*n - 258)*x^2 + (12*n^3 - 174*n^2 + 506*n - 344)*x^3 + (24*n^2 - 166*n + 142)*x^4 + (12*n - 12)*x^5)*v[i - 1] + ((18*n^2 - 18*n - 4)*x - (134*n^2 - 190*n + 8)*x^2 + (74*n^2 - 482*n + 316)*x^3 - (8*n^2 - 176*n + 520)*x^4 - (16*n - 124)*x^5 - 8*x^6)*deriv(v[i - 1], x))/(9*n^3 - 34*n^2 + 43*n - 18 - (13*n^3 - 66*n^2 + 137*n - 84)*x + (4*n^3 - 50*n^2 + 144*n - 98)*x^2 + (8*n^2 - 54*n + 46)*x^3 + (4*n - 4)*x^4)); 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]
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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
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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
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
veronese_section_2_0 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 …?
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Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›
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from
veronese_section_2_0 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 …?
References
- [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
- [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
- [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
- [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.