A132774 → Veronese 2 → θ+1
palindromicγ-positivenonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-6\,n + 11\,n^{2} - 6\,n^{3} + n^{4}\right)\,P_{n}(x) = \left(-22\,n x^{2} + 25\,n^{2} x^{2} - 2\,n^{3} x^{2} - n^{4} x^{2}\right)\,P_{n-4}(x) + \left(-8\,n x^{3} + 8\,n^{2} x^{3}\right)\,P_{n-4}'(x) + \left(-72\,x + 96\,n x - 30\,n^{2} x - 4\,n^{3} x + 2\,n^{4} x\right)\,P_{n-2}(x) + \left(-48\,x^{2} + 40\,n x^{2} - 8\,n^{2} x^{2}\right)\,P_{n-2}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)
\(P_{2}(x) = 10\,x\)
\(P_{3}(x) = 12\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 2 | |||||||
| 2 | 0 | 10 | 10 | ||||||
| 3 | 0 | 12 | 12 | ||||||
| 4 | 0 | 0 | 27 | 27 | |||||
| 5 | 0 | 0 | 30 | 30 | |||||
| 6 | 0 | 0 | 0 | 52 | 52 | ||||
| 7 | 0 | 0 | 0 | 56 | 56 | ||||
| 8 | 0 | 0 | 0 | 0 | 85 | 85 | |||
| 9 | 0 | 0 | 0 | 0 | 90 | 90 | |||
| 10 | 0 | 0 | 0 | 0 | 0 | 126 | 126 | ||
| 11 | 0 | 0 | 0 | 0 | 0 | 132 | 132 | ||
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 175 | 175 | |
| 13 | 0 | 0 | 0 | 0 | 0 | 0 | 182 | 182 | |
| 14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 232 | 232 |
Compute more
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Technical details
- Node id
n-991280066819b911· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-edf6954a4cd8adfc
Observed on the stored rows ?
- Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
- Palindromic rows: yes
- γ-positive (gamma_positive): yes
- 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 ExampleTriangle begins: 0: 1; 1: 2; 2: 0, 10; 3: 0, 12; 4: 0, 0, 27; 5: 0, 0, 30; 6: 0, 0, 0, 52; 7: 0, 0, 0, 56; 8: 0, 0, 0, 0, 85; 9: 0, 0, 0, 0, 90;
%F FormulaThe row polynomials satisfy (n^4-6*n^3+11*n^2-6*n)*P(n,x) = -(n^4+2*n^3-25*n^2+22*n)*x^2*P(n-4,x) + (8*n^2-8*n)*x^3*d/dx P(n-4,x) + (2*n^4-4*n^3-30*n^2+96*n-72)*x*P(n-2,x) - (8*n^2-40*n+48)*x^2*d/dx P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 10*x, P(3,x) = 12*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2;
P[2, x_] := 10*x;
P[3, x_] := 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 + 2*n^3 - 25*n^2 + 22*n)*x^2*P[n - 4, x] + (8*n^2 - 8*n)*x^3*D[P[n - 4, x], x] + (2*n^4 - 4*n^3 - 30*n^2 + 96*n - 72)*x*P[n - 2, x] - (8*n^2 - 40*n + 48)*x^2*D[P[n - 2, x], x])/(n^4 - 6*n^3 + 11*n^2 - 6*n)]];
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; v[3] = 10*x; v[4] = 12*x; for(n = 4, N, my(i = n + 1); v[i] = (-(n^4 + 2*n^3 - 25*n^2 + 22*n)*x^2*v[i - 4] + (8*n^2 - 8*n)*x^3*deriv(v[i - 4], x) + (2*n^4 - 4*n^3 - 30*n^2 + 96*n - 72)*x*v[i - 2] - (8*n^2 - 40*n + 48)*x^2*deriv(v[i - 2], x))/(n^4 - 6*n^3 + 11*n^2 - 6*n)); 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 multiplier sequences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Proof chains ? JSON
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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from
veronese_section_2_0 of A natural number operator.?
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Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›
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from
A natural number operator.?
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
A natural number operator.?
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from
veronese_section_2_0 of A natural number operator.?
Related ?
Same fingerprint ?
- basis-map coefficients of A026729, shifted → reverse
- veronese parent of Table read by rows: 1 if x = y, 0 otherwise, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...
- A136521: Triangle read by rows: (1, 2, 2, 2, ...) on the main diagonal and the rest zeros.
- A131436 → (θ+1)²
- A127648 → (θ+1)² → θ+1
- basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
- A196347 → γ
- A073423: Sums of two powers of zero: triangle read by rows: T(m,n) = 0^n + 0^m, n >= 0, m = 0..n.
- A132774 → Veronese 2 → D
- A130123 → θ+1
- A132440 → θ+1 → (θ+1)²
- A038338 → γ
- A133080 → Veronese 2
- basis-map coefficients of A030528, shifted → (θ+1)²
- A130123 → reverse
- basis-map coefficients of A297191, shifted → (θ+1)² → D
- basis-map coefficients of A030528, shifted → D → (θ+1)²
- A198954: Expansion of the rotational partition function for a heteronuclear diatomic molecule.
- A127733: Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.
- A132440 → θ+1 → D
- basis-map coefficients of A021012 → θ+1
- A131431 → θ+1 → reverse
- basis-map coefficients of A021012 → D
- A129184: Shift operator, right.
- A131431 → (θ+1)² → reverse
- Hadamard-factor parent of A127773 → (θ+1)²
- A356546 → γ
- A130123 → (θ+1)² → reverse
- A136572 → (θ+1)² → reverse
- A127648 → (θ+1)² → D
- A127648: Triangle read by rows: row n consists of n zeros followed by n+1.
- basis-map coefficients of A026729, shifted → θ+1 → D
- A127648 → D → θ+1
- basis-map coefficients of A112227 → D
- A131431 → (θ+1)²
- A136572 → (θ+1)² → D
- image under θ+1 → (θ+1)² → θ+1
- A038325 → γ
- A136572 → (θ+1)² → θ+1
- basis-map coefficients of A030528, shifted → θ+1 → D
- A132774 → Veronese 2
- A127773 → reverse
- A136572 → D → (θ+1)²
- A130123 → (θ+1)² → D
- basis-map coefficients of A112227 → (θ+1)² → reverse
- A127773 → (θ+1)²
- A130123 → D → D
- basis-map coefficients of A112227 → θ+1
- basis-map coefficients of A030528, shifted → θ+1
- basis-map coefficients of A026729 → D
- A131436 → D
- basis-map coefficients of A112227 → (θ+1)² → θ+1
- image under θ+1 → (θ+1)²
- A127648 → D
- A132774 → Veronese 2 (j=1) → D
- A136521 → D
- A130123 → θ+1 → D
- basis-map coefficients of A026729 → D → D
- A136572 → θ+1
- A131431 → (θ+1)² → D
- basis-map coefficients of A026729 → (θ+1)²
- A131431 → D → D
- basis-map coefficients of A021012 → reverse
- basis-map coefficients of A026729 → (θ+1)² → (θ+1)²
- basis-map coefficients of A026729 → (θ+1)² → reverse
- A127648 → (θ+1)² → (θ+1)²
- basis-map coefficients of A026729, shifted → (θ+1)² → reverse
- basis-map coefficients of A026729, shifted → (θ+1)² → (θ+1)²
- A130123 → D → (θ+1)²
- Hadamard-factor parent of A127773 → reverse
- A135838 → γ
- A136572 → θ+1 → D
- basis-map coefficients of Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.
- A130123 → D → θ+1
- basis-map coefficients of A297191, shifted → (θ+1)² → reverse
- basis-map coefficients of A112227 → θ+1 → reverse
- A127773 → θ+1 → reverse
- Hadamard-factor parent of A127773 → D
- A130123 → D
- basis-map coefficients of A026729 → (θ+1)² → D
- basis-map coefficients of A030528, shifted → θ+1 → reverse
- basis-map coefficients of A297191, shifted → (θ+1)²
- A127648 → (θ+1)²
- basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
- basis-map coefficients of A297191, shifted → D → (θ+1)²
- A023531: a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
- A131431 → D
- basis-map coefficients of A297191, shifted → D
- basis-map coefficients of A030528, shifted → D → θ+1
- basis-map coefficients of A030528, shifted → (θ+1)² → θ+1
- A131431: 3n + 1 preceded by n zeros.
- hadamard_factor parent of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
- basis-map coefficients of A297191, shifted → D → θ+1
- A003506 → γ
- image under θ+1 → θ+1 → D → D
- basis-map coefficients of A021012 → (θ+1)²
- A156992 → γ
- A136572 → D → θ+1
- basis-map coefficients of A021012 → θ+1 → (θ+1)²
- basis-map coefficients of A297191, shifted → (θ+1)² → (θ+1)²
- basis-map coefficients of A026729, shifted → D → (θ+1)²
- A127648 → D → D
- A344565 → γ
- A028326 → γ
- basis-map coefficients of A026729 → D → (θ+1)²
- A131431 → reverse
- A136572 → D
- basis-map coefficients of A297191, shifted → θ+1
- A130123 → (θ+1)² → (θ+1)²
- basis-map coefficients of A112227 → (θ+1)² → (θ+1)²
- A130123 → θ+1 → reverse
- A127773 → D → D
- image under θ+1 → (θ+1)² → reverse
- basis-map coefficients of A026729, shifted → θ+1
- A127773 → θ+1 → D
- A131431 → (θ+1)² → θ+1
- A178822 → γ
- basis-map coefficients of A026729, shifted → (θ+1)² → D
- image under θ+1 → θ+1 → (θ+1)²
- A132774 → Veronese 2 → (θ+1)²
- A136572 → D → D
- basis-map coefficients of A026729, shifted → (θ+1)² → θ+1
- A136572 → (θ+1)² → (θ+1)²
- basis-map coefficients of A112227 → θ+1 → D
- A131431 → D → θ+1
- A038221 → γ
- basis-map coefficients of A297191, shifted → (θ+1)² → θ+1
- A132774 → Veronese 2 (j=1) → (θ+1)²
- basis-map coefficients of A112227 → (θ+1)² → D
- A136572: Triangle read by rows: row n consists of n zeros followed by n!.
- A178821 → γ
- A129186: Right shift operator generating 1's in shifted spaces.
- A073424: Triangle read by rows: T(m,n) = parity of 0^n + 0^m, n = 0,1,2,3 ..., m = 0,1,2,3, ... n.
- basis-map coefficients of A026729, shifted → D → D
- image under θ+1 → D
- basis-map coefficients of A021012 → θ+1 → D
- basis-map coefficients of A026729, shifted → D → θ+1
- basis-map coefficients of A030528, shifted → reverse
- A127773 → D → θ+1
- A131431 → θ+1
- A127773 → D
- A131436 → θ+1
- basis-map coefficients of A026729 → (θ+1)² → θ+1
- A085881 → γ
- A127648 → D → (θ+1)²
- A127773 → θ+1
- basis-map coefficients of A030528, shifted → D → D
- A134309 → D
- basis-map coefficients of A112227 → (θ+1)²
- A127773: Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
- basis-map coefficients of A030528, shifted → (θ+1)² → (θ+1)²
- basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
- A130123 → (θ+1)² → θ+1
- Hadamard-factor parent of A127773 → θ+1
- basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
- image under θ+1 → θ+1 → D
- theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
- A127773 → (θ+1)² → reverse
- A130123 → (θ+1)²
- image under θ+1 → reverse
- A132440 → θ+1
- basis-map coefficients of A030528, shifted → D
- image under θ+1 → θ+1 → reverse
- A010054: a(n) = 1 if n is a triangular number, otherwise 0.
- basis-map coefficients of A030528, shifted → (θ+1)² → reverse
- basis-map coefficients of A026729 → D → θ+1
- image under θ+1 → θ+1 → D → θ+1
- basis-map coefficients of A030528, shifted → (θ+1)² → D
- A178820 → γ
- A132774 → Veronese 2 (j=1) → θ+1
- A130123: Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros …
- A131436: Triangle read by rows, (n-1) zeros followed by 2^n - 1.
- A131431 → θ+1 → D
- basis-map coefficients of A026729, shifted → θ+1 → reverse
- basis-map coefficients of A026729, shifted → D
- A131431 → (θ+1)² → (θ+1)²
- A131436 → reverse
- image under θ+1 → (θ+1)² → D
- A132440: Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the clas…
- A132774 → Veronese 2 (j=1)
- basis-map coefficients of A297191, shifted → D → D
- A134309: Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).
- A136572 → (θ+1)²
- basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
- basis-map coefficients of A scaled Hermite triangle.
- basis-map coefficients of A026729, shifted → (θ+1)²
- basis-map coefficients of A297191, shifted → θ+1 → D
- basis-map coefficients of A021012 → θ+1 → reverse
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
- [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.