{"id":"n-8fe4e03db036ba8f","revision":0,"first_index":0,"fingerprint":"f6-5520686a41bcab10","name":"theta_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["196","336","108"],["2744","7056","4536","864"],["38416","131712","127008","48384","6480"],["537824","2304960","2963520","1693440","453600","46656"],["7529536","38723328","62233920","47416320","19051200","3919104","326592"],["105413504","632481024","1219784832","1161699840","622339200","192036096","32006016","2239488"],["1475789056","10119696384","22769316864","26022076416","17425497600","7169347584","1792336896","250822656","15116544"],["20661046784","159385218048","409847703552","546463604736","439122539520","225834448896","75278149632","15801827328","1904684544","100776960"],["289254654976","2479325614080","7172334812160","10929272094720","10246192588800","6323364569088","2634735237120","737418608640","133327918080","14108774400","665127936"],["4049565169664","38181614456832","122726617896960","210388487823360","225416236953600","162299690606592","81149845303296","28390616432640","6844166461440","1086375628800","102429702144","4353564672"],["56693912375296","583137384431616","2061807180668928","3927251772702720","4733740976025600","3895192574558208","2272195668492288","953924712136704","287454991380480","60837035212800","8604094980096","731398864896","28298170368"],["793714773254144","8844250330546176","34113536989249536","71475982263189504","95726761959628800","88615631071199232","59077087380799488","28935716268146688","10463361686249472","2768085102182400","521981762125824","66557296705536","5150267006976","182849716224"],["11112006825558016","133344081906696192","557187770824409088","1273572047598649344","1876244534408724480","1929851521106116608","1447388640829587456","810200055508107264","341803148417482752","108508936005550080","25577106344165376","4348410051428352","504726166683648","35838544379904","1175462461440"],["155568095557812224","2000161228600442880","9000725528701992960","22287510832976363520","35819213838712012800","40526881943228448768","33772401619357040640","21267751457087815680","10254094452524482560","3797812760194252800","1074238466454945792","228291527699988480","35330831667855360","3763047159889920","246847116902400","7522959753216"],["2177953337809371136","29869074347099947008","144011608459231887360","384030955891285032960","668625324989290905600","825274686843924774912","756501796273597710336","529330702931963412480","287114644670685511680","121530008326216089600","40104902747651309568","10227460440959483904","1978526573399900160","280974187938447360","27646877093068800","1685142984720384","47958868426752"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["640","15200","352000","8000000","179200000","3968000000","87040000000","1894400000000","40960000000000","880640000000000","18841600000000000","401408000000000000","8519680000000000000","180224000000000000000","3801088000000000000000"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["196","336","108"]],"recurrence":{"denominator":{"coeffs":[["7","6"],["0","3"]]},"terms":[{"coeff":{"coeffs":[["98","168","54"],["0","42","18"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193292,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-eff13ee65d621e13"]},{"id":193293,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-eff13ee65d621e13"]},{"id":193294,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-eff13ee65d621e13"]},{"id":193295,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-eff13ee65d621e13"]}],"region_exact":{},"incoming_edges":[{"id":"e-eff13ee65d621e13","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-8fe4e03db036ba8f","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"0ac68cd8a04245af6d6f5172255db542b56b166048aaa09090c09f1bef1640be","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-7288f584ad081a07","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-39b00cbcc35daf27","kind":"fingerprint_related","node_a_id":"n-26b9fec6ac41e789","node_b_id":"n-8fe4e03db036ba8f","fingerprint":"f6-5520686a41bcab10","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5520686a41bcab10"},"other_id":"n-26b9fec6ac41e789","other":{"id":"n-26b9fec6ac41e789","name":"theta_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-5520686a41bcab10","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["21","18"],["441","756","243"]],"card_rows":[["1"],["21","18"],["441","756","243"],["9261","23814","15309","2916"]],"row_sums_preview":["1","39","1440","51300","1782000","60750000","2041200000","67797000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-62ade03cd9f25f04","e-8ad0953b93e301ab"],"text":"A038335 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A038335 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}},{"id":"l-39b390ac9f00d0c1","kind":"fingerprint_related","node_a_id":"n-8d0909a4318144f1","node_b_id":"n-8fe4e03db036ba8f","fingerprint":"f6-5520686a41bcab10","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5520686a41bcab10"},"other_id":"n-8d0909a4318144f1","other":{"id":"n-8d0909a4318144f1","name":"theta_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","first_index":0,"fingerprint":"f6-5520686a41bcab10","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["7","6"],["49","84","27"]],"card_rows":[["1"],["7","6"],["49","84","27"],["343","882","567","108"]],"row_sums_preview":["1","13","160","1900","22000","250000","2800000","31000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["basis A013610","\u03b8+1"],"edges":["e-b9cdb58d54bdc029","e-24454303c8f5ce1c"],"text":"A038255 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A038255 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:           196,            336,             108;\n  1:          2744,           7056,            4536,             864;\n  2:         38416,         131712,          127008,           48384,            6480;\n  3:        537824,        2304960,         2963520,         1693440,          453600,           46656;\n  4:       7529536,       38723328,        62233920,        47416320,        19051200,         3919104,         326592;\n  5:     105413504,      632481024,      1219784832,      1161699840,       622339200,       192036096,       32006016,        2239488;\n  6:    1475789056,    10119696384,     22769316864,     26022076416,     17425497600,      7169347584,     1792336896,      250822656,      15116544;\n  7:   20661046784,   159385218048,    409847703552,    546463604736,    439122539520,    225834448896,    75278149632,    15801827328,    1904684544,     100776960;\n  8:  289254654976,  2479325614080,   7172334812160,  10929272094720,  10246192588800,   6323364569088,  2634735237120,   737418608640,  133327918080,   14108774400,    665127936;\n  9: 4049565169664, 38181614456832, 122726617896960, 210388487823360, 225416236953600, 162299690606592, 81149845303296, 28390616432640, 6844166461440, 1086375628800, 102429702144, 4353564672;\n","recurrence_sentence":"The row polynomials satisfy (7+(3*n+6)*x)*P(n,x) = (98+(42*n+168)*x+(18*n+54)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 336*x + 108*x^2.","mathematica":"P[0, x_] := 196 + 336*x + 108*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((98 + (42*n + 168)*x + (18*n + 54)*x^2)*P[n - 1, x])/(7 + (3*n + 6)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 196 + 336*x + 108*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((98 + (42*n + 168)*x + (18*n + 54)*x^2)*v[i - 1])/(7 + (3*n + 6)*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A007318","\u03b8+1"],"edges":["e-37fb8ca4839f71ce","e-eff13ee65d621e13"],"text":"A038284 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A038284 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}