{"id":"n-4f94f81718ae3273","revision":0,"first_index":1,"fingerprint":"f6-ad16784f8d87fb89","name":"Matrix product of Stirling2-triangle A008277(n,k) and unsigned Lah-triangle |A008297(n,k)|.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["13","9","1"],["75","79","18","1"],["541","765","265","30","1"],["4683","8311","3870","665","45","1"],["47293","100989","59101","13650","1400","63","1"],["545835","1362439","960498","278901","38430","2618","84","1"],["7087261","20246445","16700545","5844510","1012431","92610","4494","108","1"],["102247563","328972471","310819590","127331905","26587575","3056907","199080","7230","135","1"],["1622632573","5805917469","6182314501","2902305450","709559950","98334621","8047347","392040","11055","165","1"],["28091567595","110645911399","131102078778","69392377901","19445781960","3155560276","311237388","19048227","720225","16225","198","1"],["526858348381","2265191981325","2955981690025","1741641891990","550412723861","102389807520","11824946848","872107236","41414802","1250535","23023","234","1"],["10641342970443","49589790516631","70668432599310","45872193348345","16141221201105","3386724794453","448389701760","38719252000","2215403190","83995912","2072070","31759","273","1"],["230283190977853","1156201277261949","1786538324270701","1266783600823650","491193999032100","114767002449099","17139464755413","1696669809120","113654402290","5191135950","160758598","3300570","42770","315","1"],["5315654681981355","28605950745797959","47637852615465858","36635701495438501","15520713733251090","3996715082570574","664722780934212","74225774104053","5699743141950","304755372350","11367520164","292923358","5083260","56420","360","1"],["130370767029135901","748605590542359405","1336603460175514705","1108094819890771470","509249312448439291","143298298817995230","26269180181842962","3266904371872284","282974805959703","17343258748530","757215304274","23498738172","511707378","7604100","73100","408","1"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["23","173","1602","17575","222497","3188806","50988405","899222457","17329515172","362164300173","8155216185781","196789115887252","5064722539020379","138457553073641465","4006059432756066914"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A088729","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["13","9","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["4","3"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169969,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-b5e82877f0afaeb2"]},{"id":169970,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-b5e82877f0afaeb2"]},{"id":169971,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-b5e82877f0afaeb2"]},{"id":169972,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2","region_exact":null,"edges":["e-b5e82877f0afaeb2"]}],"region_exact":{},"incoming_edges":[{"id":"e-b5e82877f0afaeb2","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-4f94f81718ae3273","premise_count":0,"theorem":"Iterated real-rootedness preservers (induction on n): if P_m is nonzero and real-rooted with every root in a closed region R and, for every n > m, P_n = a_n(x) S_n(P_(n-1)) where S_n maps every polynomial that is zero or real-rooted with every root in R to zero or a real-rooted polynomial with every root in R and a_n is real-rooted with every root in R, then every P_n is real-rooted with every root in R (the roots of a product are the roots of its factors); an exact degree step keeps every row nonzero; S_n = A + B D + C D^2 with A = 3 + x, B = 4 + 3 x, C = 2 x: Branden's Weyl step: T = A + B D + C D^2 maps every real-rooted polynomial to a real-rooted polynomial when (i) B^2 - 4 A C >= 0 on the real line, (ii) A strictly interlaces B (deg B = deg A + 1) or strictly alternates left of B (deg B = deg A), with leading coefficients of the same sign when deg A >= 1, and (iii) T(x^k) has degree deg A + k with leading coefficients of one sign for every k >= 0 (P. Branden, Linear transformations preserving the Polya frequency property, Trans. Amer. Math. Soc. 358 (2006), Corollary 3.5); (i) is necessary for a real-rootedness preserver (J. Borcea and P. Branden, Proc. London Math. Soc. 101 (2010), Theorem 1.3); with coefficients of one sign T also keeps every root in (-inf, 0]; a linear operator preserving real-rootedness preserves the alternation of two polynomials (P. Branden, Trans. Amer. Math. Soc. 358 (2006), Corollary 2.5)","lean_route":null,"lean_declarations":[],"certificate_hash":"c6f3d7f060c6c2dba63123ddf8aa01a1f59a0458a7100a5b7eb2d65e83345b14","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-f81bdde07b5d0b04","kind":"oeis_match","node_a_id":"n-4f94f81718ae3273","node_b_id":null,"fingerprint":null,"oeis_id":"A075729","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A075729","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A075729","relation":"match_row_sums"},{"oeis_id":"A088729","relation":"tail"}],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:          13,            9,            1;\n   2:          75,           79,           18,           1;\n   3:         541,          765,          265,          30,           1;\n   4:        4683,         8311,         3870,         665,          45,          1;\n   5:       47293,       100989,        59101,       13650,        1400,         63,         1;\n   6:      545835,      1362439,       960498,      278901,       38430,       2618,        84,        1;\n   7:     7087261,     20246445,     16700545,     5844510,     1012431,      92610,      4494,      108,      1;\n   8:   102247563,    328972471,    310819590,   127331905,    26587575,    3056907,    199080,     7230,    135,     1;\n   9:  1622632573,   5805917469,   6182314501,  2902305450,   709559950,   98334621,   8047347,   392040,  11055,   165,   1;\n  10: 28091567595, 110645911399, 131102078778, 69392377901, 19445781960, 3155560276, 311237388, 19048227, 720225, 16225, 198, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) + (4+3*x)*d/dx P(n-1,x) + 2*x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 13 + 9*x + x^2.","mathematica":"P[1, x_] := 13 + 9*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] + (4 + 3*x)*D[P[n - 1, x], x] + 2*x*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 13 + 9*x + x^2; for(n = 2, N, my(i = n); v[i] = (3 + x)*v[i - 1] + (4 + 3*x)*deriv(v[i - 1], x) + 2*x*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-4f94f81718ae3273","steps":[],"edges":[],"text":"A088729"},"headline":"Matrix product of Stirling2-triangle A008277(n,k) and unsigned Lah-triangle |A008297(n,k)|.","chain":null}