A013620 → D
nonnegative coefficients only rational roots
Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means
Recurrence
\(\left(-1 + n\right)\,P_{n}(x) = \left(2\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | |||||||
| 1 | 3 | 3 | ||||||
| 2 | 12 | 18 | 30 | |||||
| 3 | 36 | 108 | 81 | 225 | ||||
| 4 | 96 | 432 | 648 | 324 | 1500 | |||
| 5 | 240 | 1440 | 3240 | 3240 | 1215 | 9375 | ||
| 6 | 576 | 4320 | 12960 | 19440 | 14580 | 4374 | 56250 | |
| 7 | 1344 | 12096 | 45360 | 90720 | 102060 | 61236 | 15309 | 328125 |
Computed in your browser ?
Technical details
- Node id
n-74c81a9d12bf4d45· revision 1 · JSON- Definition source
derived:derivative- Fingerprint
f6-f06e51c139cfae05
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: 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): yes
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 ExampleTriangle begins: 0: 0; 1: 3; 2: 12, 18; 3: 36, 108, 81; 4: 96, 432, 648, 324; 5: 240, 1440, 3240, 3240, 1215; 6: 576, 4320, 12960, 19440, 14580, 4374; 7: 1344, 12096, 45360, 90720, 102060, 61236, 15309; 8: 3072, 32256, 145152, 362880, 544320, 489888, 244944, 52488; 9: 6912, 82944, 435456, 1306368, 2449440, 2939328, 2204496, 944784, 177147;
%F FormulaThe row polynomials satisfy (n-1)*P(n,x) = (2*n+3*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 3*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 3*n*x)*v[i - 1])/(n - 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)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-52ab624f3105e737 -
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-52ab624f3105e737 -
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-52ab624f3105e737 -
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-52ab624f3105e737
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-52ab624f3105e737Laguerre’s theorem on differential operators [SP14] polytool ?
-
premise (sequence):
Triangle of coefficients in expansion of (2+3x)^n.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
theorem_check liu_wang
e-0ecfea69e82c3845root: theorem checkLiu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
-
parent_provers periodic_product
e-31e5a5a6b5727160root: theorem checkParent-row provers of polytool polytool ?
-
preserver_image reversal, shift 0
e-cb393d3715764915Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
theorem_check liu_wang
e-1bc008755d2fc1a6root: theorem checkLiu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
-
basis_map A007318
e-01e7c46c78122cbcGeometric basis map (elementary) polytool ?
-
premise (sequence):
Triangle of coefficients in expansion of (1+2*x)^n.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
theorem_check liu_wang
e-a6866143cc223753root: theorem checkLiu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
-
basis_map A013609
e-41b2234a78052e4fGeometric basis map (elementary) polytool ?
-
premise (sequence):
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
veronese_section r=2, j=0, shift 0
e-292206ad4ae86d2dVeronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?
-
premise (sequence):
basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
parent_provers periodic_product
e-1991a148b9538b93root: theorem checkParent-row provers of polytool polytool ?
-
preserver_image mobius, shift 0
e-12303ad0e0b9f8ebAffine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
premise (sequence):
basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
-
-
premise (sequence):
basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
-
parent_provers periodic_product
e-b7eca039219a7f53root: theorem checkParent-row provers of polytool polytool ?
-
preserver_image mobius, shift 0
e-7e44b66e04c497d6Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
premise (sequence):
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
-
-
premise (sequence):
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
-
gamma_polynomial scale 1, shift 0
e-4c4f74d9ba733cffGamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
-
premise (sequence):
Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
parent_provers periodic_product
e-eab26daf7024ea74root: theorem checkParent-row provers of polytool polytool ?
-
-
premise (sequence):
Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
-
preserver_image reversal, shift 0
e-6d15baf5e54583eeAffine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
theorem_check liu_wang
e-0d085d05819c04f2root: theorem checkLiu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
-
preserver_image reversal, shift 0
e-045f9d56b54ceaf2Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
Triangle of coefficients in expansion of (1+2*x)^n.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
↵ theorem_check liu_wang
e-a6866143cc223753(expanded above) -
↵ basis_map A013609
e-41b2234a78052e4f(expanded above) -
↵ gamma_polynomial scale 1, shift 0
e-4c4f74d9ba733cff(expanded above) -
parent_provers periodic_product
e-d08b7f9df01d4f3croot: theorem checkParent-row provers of polytool polytool ?
-
preserver_image mobius, shift 0
e-2c323256b27180b8Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
binomial
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
theorem_check liu_wang
e-317eb4ccca60c6acroot: theorem checkLiu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
-
preserver_image reversal, shift 0
e-033cf842aad3bdf7Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
binomial
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
↵ theorem_check liu_wang
e-317eb4ccca60c6ac(expanded above) -
basis_map A007318
e-095baa2a9dfe188fGeometric basis map (elementary) polytool ?
-
premise (sequence):
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
premise (sequence):
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
-
parent_provers periodic_product
e-13983862a411779croot: theorem checkParent-row provers of polytool polytool ?
-
preserver_image reversal, shift 0
e-989c95e5c893fce9Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
Riordan array (1-x,x/(1-x)).
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
parent_provers periodic_product
e-b62074ba45057e82root: theorem checkParent-row provers of polytool polytool ?
-
-
premise (sequence):
Riordan array (1-x,x/(1-x)).
-
gamma_polynomial scale 1, shift 0
e-c76adc2a99cd3deaGamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
-
premise (sequence):
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 and 0<=k<=2n.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
preserver_image reversal, shift 0
e-0e058c428dfdfeb6Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
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 and 0<=k<=2n.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
parent_provers periodic_product
e-d82c9b48e4b31a6broot: theorem checkParent-row provers of polytool polytool ?
-
-
premise (sequence):
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 and 0<=k<=2n.
-
↵ parent_provers periodic_product
e-d82c9b48e4b31a6b(expanded above)
-
-
premise (sequence):
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 and 0<=k<=2n.
-
preserver_image affine, shift 0
e-3a52de7742ed9314Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
premise (sequence):
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
-
↵ theorem_check liu_wang
-
premise (sequence):
binomial
-
↵ basis_map A007318
e-095baa2a9dfe188f(expanded above) -
↵ parent_provers periodic_product
e-13983862a411779c(expanded above) -
↵ preserver_image reversal, shift 0
e-989c95e5c893fce9(expanded above) -
↵ gamma_polynomial scale 1, shift 0
e-c76adc2a99cd3dea(expanded above) -
↵ preserver_image affine, shift 0
e-3a52de7742ed9314(expanded above)
-
-
premise (sequence):
binomial
-
↵ theorem_check liu_wang
-
premise (sequence):
Triangle of coefficients in expansion of (1+2*x)^n.
-
parent_provers periodic_product
e-1697ba43e31b7729root: theorem checkParent-row provers of polytool polytool ?
-
basis_map A007318
e-4608a916413bc814Geometric basis map (elementary) polytool ?
-
premise (sequence):
binomial
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
↵ theorem_check liu_wang
e-317eb4ccca60c6ac(expanded above) -
↵ preserver_image reversal, shift 0
e-033cf842aad3bdf7(expanded above) -
↵ basis_map A007318
e-095baa2a9dfe188f(expanded above) -
↵ parent_provers periodic_product
e-13983862a411779c(expanded above) -
↵ preserver_image reversal, shift 0
e-989c95e5c893fce9(expanded above) -
↵ gamma_polynomial scale 1, shift 0
e-c76adc2a99cd3dea(expanded above) -
↵ preserver_image affine, shift 0
e-3a52de7742ed9314(expanded above)
-
↵ theorem_check liu_wang
-
premise (sequence):
binomial
-
preserver_image affine, shift 0
e-17c52917511dc051Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
binomial
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
↵ theorem_check liu_wang
e-317eb4ccca60c6ac(expanded above) -
↵ preserver_image reversal, shift 0
e-033cf842aad3bdf7(expanded above) -
↵ basis_map A007318
e-095baa2a9dfe188f(expanded above) -
↵ parent_provers periodic_product
e-13983862a411779c(expanded above) -
↵ preserver_image reversal, shift 0
e-989c95e5c893fce9(expanded above) -
↵ gamma_polynomial scale 1, shift 0
e-c76adc2a99cd3dea(expanded above) -
↵ preserver_image affine, shift 0
e-3a52de7742ed9314(expanded above)
-
↵ theorem_check liu_wang
-
premise (sequence):
binomial
-
-
premise (sequence):
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
-
↵ parent_provers periodic_product
e-d08b7f9df01d4f3c(expanded above) -
↵ preserver_image mobius, shift 0
e-2c323256b27180b8(expanded above)
-
-
premise (sequence):
Triangle of coefficients in expansion of (1+2*x)^n.
-
basis_map A013609
e-715b5f861d0b7004Geometric basis map (elementary) polytool ?
-
premise (sequence):
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
↵ theorem_check liu_wang
e-0d085d05819c04f2(expanded above) -
↵ preserver_image reversal, shift 0
e-045f9d56b54ceaf2(expanded above) -
↵ parent_provers periodic_product
e-1697ba43e31b7729(expanded above) -
↵ basis_map A007318
e-4608a916413bc814(expanded above) -
↵ preserver_image affine, shift 0
e-17c52917511dc051(expanded above)
-
↵ theorem_check liu_wang
-
premise (sequence):
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
-
parent_provers periodic_product
e-71e0b3707c4ee7dbroot: theorem checkParent-row provers of polytool polytool ?
-
preserver_image reversal, shift 0
e-dc7bfe74658d97caAffine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
Triangle of coefficients in expansion of (2+3x)^n.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
↵ theorem_check liu_wang
e-0ecfea69e82c3845(expanded above) -
↵ parent_provers periodic_product
e-31e5a5a6b5727160(expanded above) -
basis_map A013610
e-e9dcc990956c90e9Geometric basis map (elementary) polytool ?
-
premise (sequence):
binomial
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
↵ theorem_check liu_wang
e-317eb4ccca60c6ac(expanded above) -
↵ preserver_image reversal, shift 0
e-033cf842aad3bdf7(expanded above) -
↵ basis_map A007318
e-095baa2a9dfe188f(expanded above) -
↵ parent_provers periodic_product
e-13983862a411779c(expanded above) -
↵ preserver_image reversal, shift 0
e-989c95e5c893fce9(expanded above) -
↵ gamma_polynomial scale 1, shift 0
e-c76adc2a99cd3dea(expanded above) -
↵ preserver_image affine, shift 0
e-3a52de7742ed9314(expanded above)
-
↵ theorem_check liu_wang
-
premise (sequence):
binomial
-
preserver_image mobius, shift 0
e-57079e0e5fed1718Affine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
↵ theorem_check liu_wang
e-0d085d05819c04f2(expanded above) -
↵ preserver_image reversal, shift 0
e-045f9d56b54ceaf2(expanded above) -
↵ parent_provers periodic_product
e-1697ba43e31b7729(expanded above) -
↵ basis_map A007318
e-4608a916413bc814(expanded above) -
↵ preserver_image affine, shift 0
e-17c52917511dc051(expanded above)
-
↵ theorem_check liu_wang
-
premise (sequence):
Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
-
↵ theorem_check liu_wang
-
premise (sequence):
Triangle of coefficients in expansion of (2+3x)^n.
-
preserver_image affine, shift 0
e-2582f481aa50840bAffine, reversal and Möbius substitutions (elementary) polytool ?
-
premise (sequence):
Triangle of coefficients in expansion of (1+2*x)^n.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
-
↵ theorem_check liu_wang
e-a6866143cc223753(expanded above) -
↵ basis_map A013609
e-41b2234a78052e4f(expanded above) -
↵ gamma_polynomial scale 1, shift 0
e-4c4f74d9ba733cff(expanded above) -
↵ preserver_image reversal, shift 0
e-6d15baf5e54583ee(expanded above) -
↵ parent_provers periodic_product
e-d08b7f9df01d4f3c(expanded above) -
↵ preserver_image mobius, shift 0
e-2c323256b27180b8(expanded above)
-
↵ theorem_check liu_wang
-
premise (sequence):
Triangle of coefficients in expansion of (1+2*x)^n.
-
-
premise (sequence):
Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
-
↵ basis_map A013610
e-e9dcc990956c90e9(expanded above) -
↵ preserver_image mobius, shift 0
e-57079e0e5fed1718(expanded above)
-
-
premise (sequence):
Triangle of coefficients in expansion of (2+3x)^n.
Related sequences ?
Same fingerprint (23)
Equal up to content, sign, reversal and a small index shift.
- A178821 → basis A013610
- A344565 → basis A013610
- A120910 → x ↦ x/(x+1)
- A135838 → basis A013610
- A156992 → basis A013610
- A038325 → basis A013610
- A028326 → basis A013610
- A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
- A038221 → basis A013610
- A178820 → basis A013610
- A038338 → basis A013610
- A196347 → basis A013610
- A038208 → basis A013610
- Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
- A120910 → basis A013609
- A085880 → basis A013610
- A356546 → basis A013610
- A038220 → D
- A085881 → basis A013610
- A013620: Triangle of coefficients in expansion of (2+3x)^n.
- A038243 → basis A013611
- A178822 → basis A013610
- basis-map coefficients of A113214, shifted → basis A013610
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
- [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.