A003506 → basis A013609
palindromicγ-positivenonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(-4 - 8\,x - 4\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | |||||||
| 1 | 1 | 1 | ||||||
| 2 | 4 | 4 | 8 | |||||
| 3 | 12 | 24 | 12 | 48 | ||||
| 4 | 32 | 96 | 96 | 32 | 256 | |||
| 5 | 80 | 320 | 480 | 320 | 80 | 1280 | ||
| 6 | 192 | 960 | 1920 | 1920 | 960 | 192 | 6144 | |
| 7 | 448 | 2688 | 6720 | 8960 | 6720 | 2688 | 448 | 28672 |
Compute more
?
Technical details
- Node id
n-93a9da031f1ee988· revision 0 · JSON- Definition source
derived:basis_map_A013609- Fingerprint
f6-77a7ed3b1d5f257e
Observed on the stored rows ?
- Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
- 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: 0; 1: 1; 2: 4, 4; 3: 12, 24, 12; 4: 32, 96, 96, 32; 5: 80, 320, 480, 320, 80; 6: 192, 960, 1920, 1920, 960, 192; 7: 448, 2688, 6720, 8960, 6720, 2688, 448; 8: 1024, 7168, 21504, 35840, 35840, 21504, 7168, 1024; 9: 2304, 18432, 64512, 129024, 161280, 129024, 64512, 18432, 2304;
%F FormulaThe row polynomials satisfy P(n,x) = -(4+8*x+4*x^2)*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(4 + 8*x + 4*x^2)*P[n - 2, x] + (4 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(4 + 8*x + 4*x^2)*v[i - 2] + (4 + 4*x)*v[i - 1]); 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]
-
Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
-
Coefficients of one sign?
Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Proof chains ? JSON
-
Geometric basis map (elementary) polytool ? step ›
-
from
Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.?
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
binomial?
-
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
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
binomial?
- ↑ Liu–Wang theorem (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
-
Geometric basis map (elementary) polytool ? step ›
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Riordan array (1-x,x/(1-x)).?
-
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
-
from
Riordan array (1-x,x/(1-x)).?
-
Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
-
from
binomial?
- ↑ Geometric basis map (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Gamma criterion (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
binomial?
-
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
-
-
from
Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.?
-
Geometric basis map (elementary) polytool ? step ›
-
from
Exponential Riordan array [x*exp(x),x].?
-
Geometric basis map (elementary) polytool ? step ›
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
binomial?
- ↑ Liu–Wang theorem (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Gamma criterion (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
binomial?
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
binomial?
- ↑ Liu–Wang theorem (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Iterated real-rootedness preservers (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Gamma criterion (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
binomial?
-
-
from
Exponential Riordan array [x*exp(x),x].?
Related ?
Same fingerprint ?
- A135838 → D
- A178820: Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
- Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
- basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0
- A135089 → basis A013609
- A216973: Exponential Riordan array [x*exp(x),x].
- A028326 → basis A013609 → D
- A038325: Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.
- veronese parent of Binomial coefficients C(n,k) with n-k odd, read by rows.
- A028326 → basis A013609
- A344565 → basis A013609
- A135065: A127733 * A007318 as infinite lower triangular matrices.
- A135065 → x ↦ x+1 → basis A013610
- A127648 → (θ+1)² → x ↦ x+1
- basis-map coefficients of A030528, shifted → θ+1 → basis A007318
- A356546: Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
- basis-map coefficients of Triangle read by rows: row n consists of n-1 zeros followed by Fibonacci(n).
- A344565: Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
- A038338 → D
- A038325 → basis A013609 → basis A013609
- A085880 → D → D
- A156992 → D
- A135065 → basis A013609
- A376827 → basis A013609
- A178821 → basis A013609 → D
- A094305 → D
- A085880 → D → basis A013609
- basis-map coefficients of A113214, shifted → basis A013609 → D
- A130123 → θ+1 → basis A007318
- basis-map coefficients of A030528, shifted → (θ+1)² → x ↦ x+1
- A356546 → D
- A028326 → D → D
- basis-map coefficients of A026729, shifted → D → basis A007318
- A135089: Triangle T(n,k) = 5*binomial(n,k) with T(0,0) = 1, read by rows.
- A258758: Triangle T(n,k) = C(n+k-1,k)*C(2*n-1,n-k).
- A196347 → basis A013609 → basis A013609
- A097805: Number of compositions of n with k parts, T(n, k) = binomial(n-1, k-1) for n, k >= 1 and T(n, 0) = 0^n, triang…
- basis-map coefficients of A113214, shifted → D → basis A013609
- A038221 → D
- A134346: Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).
- A094305: Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).
- A356546 → x ↦ x+1 → basis A013610
- A007318 → ×(−(1/2)n+(1/2)n²) → D
- A038221: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.
- A356546 → basis A013609 → basis A013609
- A131431 → (θ+1)² → x ↦ x+1
- basis-map coefficients of A113214, shifted → D → D
- A134059 → basis A013609
- basis-map coefficients of A026729, shifted → (θ+1)² → basis A007318
- A132200 → basis A013609
- basis-map coefficients of A113214, shifted → basis A013609 → basis A013609
- A356546 → basis A013609 → D
- A376827: T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).
- A003506 → x ↦ x+1 → basis A013610
- Second weight array of Pascal's triangle (formatted as a rectangle), by antidiagonals.
- A085880 → x ↦ x+1 → basis A013610
- A219570: Triangular array read by rows. T(n,k) is the number of necklaces (turning over is not allowed) of n labeled b…
- A156992 → basis A013609 → D
- A038279 → basis A013610 → basis A013611
- A027465 → basis A013611 → basis A013609
- A027465 → basis A013611
- A003506 → D → D
- A038221 → x ↦ x+1 → basis A013610
- A156992 → basis A013609 → basis A013609
- basis-map coefficients of A026729 → D → basis A007318
- A124860: A Jacobsthal-Pascal triangle.
- basis-map coefficients of A026729 → basis A007318 → basis A013609
- A127773 → (θ+1)² → x ↦ x+1
- A127952: Triangle read by rows, T(n,k) = (n+1)*binomial(n-1,k-1).
- A038208: Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
- A134059: Triangle T(n, k) = 3*binomial(n,k) with T(0, 0) = 1, read by rows.
- A132200: Numbers in (4,4)-Pascal triangle .
- A178821: Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
- A135065 → D
- A131431 → θ+1 → basis A007318
- A178822 → D
- A085881: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).
- A376827 → D
- A038338 → x ↦ x+1 → basis A013610
- A178821 → basis A013609
- A196347: Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).
- A124860 → D
- A038338 → basis A013609 → D
- A178820 → basis A013609
- Hadamard-factor parent of A127773 → x ↦ x+1
- A120910 → basis A007318 → basis A013611
- A003506: Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.
- basis-map coefficients of A113214, shifted → D
- basis-map coefficients of A026729, shifted → basis A007318
- A131431 → x ↦ x+1
- A131431 → x ↦ x+1 → D
- veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).
- veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).
- basis-map coefficients of A021012 → θ+1 → basis A007318
- basis-map coefficients of A127647 → D
- basis-map coefficients of A112227 → x ↦ x+1 → basis A013609
- A085881 → D
- A124860 → basis A013609
- A127773 → θ+1 → x ↦ x+1
- A135089 → D
- A178822 → x ↦ x+1 → basis A013610
- A178821 → D
- A178820 → basis A013609 → basis A013609
- basis-map coefficients of A112227 → θ+1 → x ↦ x+1
- A038325 → basis A007318 → basis A013610
- A134346 → basis A013609
- basis-map coefficients of A026729, shifted → basis A007318 → basis A013609
- A135838 → basis A013609
- A178822 → basis A013609 → basis A013609
- A178820 → D
- A178822: Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.
- basis-map coefficients of A112227 → (θ+1)² → basis A007318
- A003506 → D
- A027465 → D → basis A013611
- A003506 → D → basis A013609
- A237765: Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.
- A258758 → basis A013609
- A038330 → basis A013611 → D
- A135838: Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
- basis-map coefficients of A113214, shifted → basis A013609
- basis-map coefficients of A113214, shifted → x ↦ x+1 → basis A013610
- A132200 → D
- A380114: Triangle read by rows: The convolution triangle of 2^n, where the convolution triangle of a sequence is define…
- A027465 → basis A013611 → D
- A156992 → basis A007318 → basis A013610
- A120910 → basis A013610 → D
- veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).
- basis-map coefficients of A127647 → basis A013609
- A028326 → basis A007318 → basis A013610
- A134058: Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.
- basis-map coefficients of A021012 → x ↦ x+1
- A136572 → (θ+1)² → basis A007318
- A134058 → basis A013609
- A196347 → basis A013609 → D
- A038325 → basis A013609
- A178820 → basis A013609 → D
- A038338 → basis A013609
- A178821 → basis A007318 → basis A013610
- A134346 → D
- A085881 → basis A013609 → D
- A085880 → D
- A094305 → x ↦ x+1 → basis A013610
- A085881 → basis A013609 → basis A013609
- Riordan array (1-x,x/(1-x)).
- basis-map coefficients of A026729 → basis A007318
- A094305 → basis A013609
- A038221 → basis A013609
- A131431 → x ↦ x+1 → basis A013609
- A178822 → basis A013609 → D
- A178821 → basis A013609 → basis A013609
- A178822 → basis A013609
- A132440 → θ+1 → x ↦ x+1
- A085880: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A00010…
- veronese parent of Binomial coefficients C(n,k) with n-k even, read by rows.
- A038208 → D → D
- A038325 → basis A013609 → D
- A196347 → x ↦ x+1 → basis A013610
- basis-map coefficients of A112227 → x ↦ x+1
- A120910 → basis A013610
- A085881 → basis A013609
- A038325 → D
- A085880 → basis A013609 → D
- A028326 → D
- A156992 → basis A013609
- A178820 → basis A007318 → basis A013610
- A129186 → basis A007318
- A134309 → basis A007318
- A038330 → basis A013611 → basis A013609
- A038221 → basis A013609 → D
- A038338 → basis A013609 → basis A013609
- A038330 → basis A013611
- A085881 → basis A007318 → basis A013610
- A134059 → D
- A038208 → D → basis A013609
- basis-map coefficients of A026729 → (θ+1)² → basis A007318
- A196347 → basis A013609
- A085880 → basis A013609
- A120910 → basis A013610 → basis A013609
- A085880 → basis A013609 → basis A013609
- A129533: Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.
- A156992: Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
- A258758 → D
- A356546 → basis A013609
- image under θ+1 → x ↦ x+1
- A038208 → D
- A007318: Binomial
- A344565 → D
- image under θ+1 → (θ+1)² → x ↦ x+1
- A028326 → basis A013609 → basis A013609
- A028326: Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).
- A196347 → D
- basis-map coefficients of A026729, shifted → θ+1 → x ↦ x+1
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.