A038291 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(10 + x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	10	1														11
2	100	21	1													122
3	1000	331	33	1												1365
4	10000	4641	727	46	1											15415
5	100000	61051	13365	1325	60	1										175802
6	1000000	771561	221431	30590	2165	75	1									2025823
7	10000000	9487171	3428733	619101	60900	3290	91	1								23599287
8	100000000	114358881	50631967	11477046	1471701	110250	4746	108	1							278054700
9	1000000000	1357947691	721942485	199833565	32080860	3125451	186186	6582	126	1						3315122947
10	10000000000	15937424601	10021257511	3319778830	648965605	78962625	6104427	298080	8850	145	1					40012800675
11	100000000000	185311670611	136192514733	53178382301	12405297300	1833404980	176633457	11171787	457380	11605	165	1				489109544320
12	1000000000000	2138428376721	1819621847407	827511484646	226852544501	39906372000	4659540292	366553836	19404627	677875	14905	186	1			6057366816997
13	10000000000000	24522712143931	23973890545605	12577271147805	4003447107660	825448124501	114459016672	10890955504	715837122	32284252	975975	18811	208	1		76028868158047
14	100000000000000	279749833583241	312209398691191	187478415467070	68625530655045	16385168975175	2656792391253	299605260240	23776023700	1329237910	51803752	1371006	23387	231	1	967429903483202

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Technical details

Node id: n-ff20a023b847453c · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-4531bfe97cc9590a

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

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 Example

Triangle begins:
  0:          1;
  1:         10,          1;
  2:        100,         21,         1;
  3:       1000,        331,        33,         1;
  4:      10000,       4641,       727,        46,        1;
  5:     100000,      61051,     13365,      1325,       60,       1;
  6:    1000000,     771561,    221431,     30590,     2165,      75,      1;
  7:   10000000,    9487171,   3428733,    619101,    60900,    3290,     91,    1;
  8:  100000000,  114358881,  50631967,  11477046,  1471701,  110250,   4746,  108,   1;
  9: 1000000000, 1357947691, 721942485, 199833565, 32080860, 3125451, 186186, 6582, 126, 1;

%F Formula

The row polynomials satisfy P(n,x) = (10+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(10 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (10 + x)*v[i - 1] + x*deriv(v[i - 1], x)); 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-f0a8a7a397577cd8
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-f0a8a7a397577cd8
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-f0a8a7a397577cd8

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

basis_map A008277 e-f0a8a7a397577cd8

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-c055c0541343b096 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-6d834ef5c24f37f2 root: theorem check
    
    Parent-row provers of polytool polytool ?
  - basis_map A007318 e-a986b288b7f6003d
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-b6a9810385e1026e root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    basis_map A007318 e-403b5075da2bf036
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*7^(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-c0e43064b314c6c0 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    parent_provers periodic_product e-7e8833b30e1c308e root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    basis_map A007318 e-cffbbf28e801bdbf
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*6^(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-29094c360cdb0314 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    parent_provers periodic_product e-e8d1c828e3ebc417 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    basis_map A007318 e-efe1634c04c9236a
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is 5^(i-j)*binomial(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-f2f89dae9298aff8 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    parent_provers periodic_product e-868f364735375874 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    basis_map A007318 e-8f220a12633250f0
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*4^(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-2d1c3ef96dc1c123 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    preserver_image reversal, shift 0 e-6e49a6f9dd41e218
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle of coefficients in expansion of (1+4x)^n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-f45969f98fa0cee8 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    parent_provers periodic_product e-36f1e280ef03dec1 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image reversal, shift 0 e-8a33d9f811421624
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*4^(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-2d1c3ef96dc1c123 (expanded above)
    
    basis_map A007318 e-9f0b70d11cb8d495
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): Cube of lower triangular normalized binomial matrix.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-2786234a932620b2 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    preserver_image reversal, shift 0 e-090c65763405a203
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle of coefficients in expansion of (1+3*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-035d18f235569c81 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    basis_map A013610 e-404109e50a21afd4
    
    Geometric 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-292206ad4ae86d2d
    
    Veronese 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-1991a148b9538b93 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image mobius, shift 0 e-12303ad0e0b9f8eb
    
    Affine, 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]
    
    ↵ parent_provers periodic_product e-1991a148b9538b93 (expanded above)
    
    parent_provers periodic_product e-b7eca039219a7f53 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image mobius, shift 0 e-7e44b66e04c497d6
    
    Affine, 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]
    
    ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
    
    ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
    
    preserver_image reversal, shift 0 e-b03bb7746bf562de
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Cube of lower triangular normalized binomial matrix.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-2786234a932620b2 (expanded above)
    
    parent_provers periodic_product e-58678fea8a24dd74 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    basis_map A007318 e-7b97c40e19745577
    
    Geometric 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 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    preserver_image reversal, shift 0 e-045f9d56b54ceaf2
    
    Affine, 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 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    basis_map A013609 e-41b2234a78052e4f
    
    Geometric 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-292206ad4ae86d2d (expanded above)
    
    ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
    
    ↵ preserver_image mobius, shift 0 e-7e44b66e04c497d6 (expanded above)
    
    gamma_polynomial scale 1, shift 0 e-4c4f74d9ba733cff
    
    Gamma 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-eab26daf7024ea74 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image reversal, shift 0 e-6d15baf5e54583ee
    
    Affine, 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)
    
    parent_provers periodic_product e-1697ba43e31b7729 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    basis_map A007318 e-4608a916413bc814
    
    Geometric 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 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    preserver_image reversal, shift 0 e-033cf842aad3bdf7
    
    Affine, 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-095baa2a9dfe188f
    
    Geometric 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-292206ad4ae86d2d (expanded above)
    
    ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
    
    ↵ preserver_image mobius, shift 0 e-7e44b66e04c497d6 (expanded above)
    
    parent_provers periodic_product e-13983862a411779c root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image reversal, shift 0 e-989c95e5c893fce9
    
    Affine, 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-b62074ba45057e82 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    gamma_polynomial scale 1, shift 0 e-c76adc2a99cd3dea
    
    Gamma 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-0e058c428dfdfeb6
    
    Affine, 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-d82c9b48e4b31a6b root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    ↵ parent_provers periodic_product e-d82c9b48e4b31a6b (expanded above)
    
    preserver_image affine, shift 0 e-3a52de7742ed9314
    
    Affine, 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]
    
    ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
    
    ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
    
    ↵ preserver_image mobius, shift 0 e-7e44b66e04c497d6 (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)
    
    preserver_image affine, shift 0 e-17c52917511dc051
    
    Affine, 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)
    
    parent_provers periodic_product e-d08b7f9df01d4f3c root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image mobius, shift 0 e-2c323256b27180b8
    
    Affine, 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)
    
    ↵ parent_provers periodic_product e-1697ba43e31b7729 (expanded above)
    
    ↵ basis_map A007318 e-4608a916413bc814 (expanded above)
    
    ↵ preserver_image affine, shift 0 e-17c52917511dc051 (expanded above)
    
    preserver_image affine, shift 0 e-4b09f87a5a93f62b
    
    Affine, 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)
    
    parent_provers periodic_product e-bd3566bc3465f71b root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image mobius, shift 0 e-33c9dd76a325511a
    
    Affine, 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)
    
    ↵ parent_provers periodic_product e-58678fea8a24dd74 (expanded above)
    
    ↵ basis_map A007318 e-7b97c40e19745577 (expanded above)
    
    ↵ preserver_image affine, shift 0 e-4b09f87a5a93f62b (expanded above)
    
    parent_provers periodic_product e-bf59fe80a0559f3e root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image affine, shift 0 e-4db64ce997502606
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Cube of lower triangular normalized binomial matrix.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-2786234a932620b2 (expanded above)
    
    ↵ preserver_image reversal, shift 0 e-090c65763405a203 (expanded above)
    
    ↵ parent_provers periodic_product e-58678fea8a24dd74 (expanded above)
    
    ↵ basis_map A007318 e-7b97c40e19745577 (expanded above)
    
    ↵ preserver_image affine, shift 0 e-4b09f87a5a93f62b (expanded above)
    
    basis_map A013611 e-c19dc1e19e56bc04
    
    Geometric 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-292206ad4ae86d2d (expanded above)
    
    ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
    
    ↵ preserver_image mobius, shift 0 e-7e44b66e04c497d6 (expanded above)
    
    preserver_image mobius, shift 0 e-db5094ba61234dec
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle of coefficients in expansion of (1+3*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-035d18f235569c81 (expanded above)
    
    ↵ basis_map A013610 e-404109e50a21afd4 (expanded above)
    
    ↵ preserver_image reversal, shift 0 e-b03bb7746bf562de (expanded above)
    
    ↵ parent_provers periodic_product e-bd3566bc3465f71b (expanded above)
    
    ↵ preserver_image mobius, shift 0 e-33c9dd76a325511a (expanded above)
    
    ↵ basis_map A007318 e-9f0b70d11cb8d495 (expanded above)
    
    ↵ parent_provers periodic_product e-bf59fe80a0559f3e (expanded above)
    
    ↵ preserver_image affine, shift 0 e-4db64ce997502606 (expanded above)
    
    preserver_image affine, shift 0 e-0ab1d5f59b4d29a0
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*4^(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-2d1c3ef96dc1c123 (expanded above)
    
    ↵ preserver_image reversal, shift 0 e-6e49a6f9dd41e218 (expanded above)
    
    ↵ basis_map A007318 e-9f0b70d11cb8d495 (expanded above)
    
    ↵ parent_provers periodic_product e-bf59fe80a0559f3e (expanded above)
    
    ↵ preserver_image affine, shift 0 e-4db64ce997502606 (expanded above)
    
    parent_provers periodic_product e-e44a3cf5ca807197 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741