{"id":"n-df3f3321552d1634","revision":0,"first_index":0,"fingerprint":"f6-697af679532967a7","name":"theta_plus_one_squared of basis_map_A013609 of Triangle of coefficients in expansion of (1+6x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["7","48"],["49","672","1296"],["343","7056","27216","27648"],["2401","65856","381024","774144","518400"],["16807","576240","4445280","13547520","18144000","8957952"],["117649","4840416","46675440","189665280","381024000","376233984","146313216"],["823543","39530064","457419312","2323399680","6223392000","9217732608","7169347584","2293235712"],["5764801","316240512","4269246912","26022076416","87127488000","172064342016","200741732352","128421199872","34828517376"],["40353607","2490394032","38423222208","273231802368","1097806348800","2710013386752","4215576379392","4045267795968","2194196594688","515978035200"],["282475249","19369731360","336203194320","2732318023680","12807740736000","37940187414528","73772586639360","94389581905920","76796880814080","36118462464000","7492001071104"],["1977326743","149146931472","2876405106960","26298560977920","140885148096000","486899071819776","1136097834246144","1816999451688960","1971119940894720","1390560804864000","576884082475008","106993205379072"],["13841287201","1138940203968","24161802898464","245453235793920","1479294055008000","5842788861837312","15905369679446016","30525590788374528","41393518758789120","38935702536192000","24229131463950336","8987429251842048","1506820975755264"],["96889010407","8636963213424","199884005796384","2233624445724672","14957306556192000","66461723303399424","206769805832798208","462971460290347008","753362041409961984","885787232698368000","734950321073160192","408928030958813184","137120708793729024","20970668254298112"],["678223072849","65109414993504","1632386047337136","19899563243728896","146581604250681600","723694320414793728","2532930121451778048","6481600444064858112","12304913343029379072","17361429760888012800","18006282866292424704","13358315677987897344","6718914730892722176","2055125488921214976","288881654523494400"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","55","2017","62263","1741825","45687799","1144869985","27724880503","653796608257","15093024313015","342405614469601","7655765629806775","169076414191378177","3694721541474812023","80000267749646389345"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["49","-84","0"],["0","252","0"],["0","0","144"]]},"terms":[{"coeff":{"coeffs":[["343","1764","3024","1728"],["0","1764","5040","3456"],["0","0","1008","1728"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190586,"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-f675d9372fd4fccf"]},{"id":190587,"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-f675d9372fd4fccf"]},{"id":190588,"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-f675d9372fd4fccf"]},{"id":190589,"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-f675d9372fd4fccf"]}],"region_exact":{},"incoming_edges":[{"id":"e-f675d9372fd4fccf","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-df3f3321552d1634","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":"82c64b36ab3c94503879ac1f791b64aec10303511f50959b5dcb986ee59fe5f7","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-96d5ef3a0d05e14f","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+6x)^n.","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":[],"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:        1;\n  1:        7,         48;\n  2:       49,        672,        1296;\n  3:      343,       7056,       27216,        27648;\n  4:     2401,      65856,      381024,       774144,        518400;\n  5:    16807,     576240,     4445280,     13547520,      18144000,       8957952;\n  6:   117649,    4840416,    46675440,    189665280,     381024000,     376233984,     146313216;\n  7:   823543,   39530064,   457419312,   2323399680,    6223392000,    9217732608,    7169347584,    2293235712;\n  8:  5764801,  316240512,  4269246912,  26022076416,   87127488000,  172064342016,  200741732352,  128421199872,   34828517376;\n  9: 40353607, 2490394032, 38423222208, 273231802368, 1097806348800, 2710013386752, 4215576379392, 4045267795968, 2194196594688, 515978035200;\n","recurrence_sentence":"The row polynomials satisfy (49+(252*n-84)*x+144*n^2*x^2)*P(n,x) = (343+(1764*n+1764)*x+(1008*n^2+5040*n+3024)*x^2+(1728*n^2+3456*n+1728)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((343 + (1764*n + 1764)*x + (1008*n^2 + 5040*n + 3024)*x^2 + (1728*n^2 + 3456*n + 1728)*x^3)*P[n - 1, x])/(49 + (252*n - 84)*x + 144*n^2*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((343 + (1764*n + 1764)*x + (1008*n^2 + 5040*n + 3024)*x^2 + (1728*n^2 + 3456*n + 1728)*x^3)*v[i - 1])/(49 + (252*n - 84)*x + 144*n^2*x^2)); 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-a6a633b989da1879","steps":["x \u21a6 x/(x+1)","(\u03b8+1)\u00b2"],"edges":["e-05b85140032a040d","e-f675d9372fd4fccf"],"text":"A038271 \u2192 x \u21a6 x/(x+1) \u2192 (\u03b8+1)\u00b2"},"headline":"A038271 \u2192 x \u21a6 x/(x+1) \u2192 (\u03b8+1)\u00b2","chain":null}