{"id":"n-7076c2900694d81e","revision":0,"first_index":0,"fingerprint":"f6-f47695d0d66fbaa8","name":"theta_plus_one of basis_map_A013610 of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["1","12","27"],["1","24","162","432","405"],["1","36","405","2160","6075","8748","5103"],["1","48","756","6048","28350","81648","142884","139968","59049"],["1","60","1215","12960","85050","367416","1071630","2099520","2657205","1968300","649539"],["1","72","1782","23760","200475","1154736","4715172","13856832","29229255","43302600","42869574","25509168","6908733"],["1","84","2457","39312","405405","2918916","15324309","60046272","177324147","394053660","650188539","773778096","628694703","312487308","71744535"],["1","96","3240","60480","737100","6368544","40864824","200154240","759960630","2251735200","5201508312","9285337152","12573894060","12499492320","8609344200","3673320192","731794257"],["1","108","4131","88128","1239300","12492144","94732092","556792704","2583866142","9569874600","28422527562","67650313536","128253719412","191242232496","219538277100","187339329792","111964521321","41841412812","7360989291"],["1","120","5130","123120","1962225","22604832","197792280","1356289920","7438402530","33059566800","120006227484","357043321440","870293096010","1730286865440","2780818176600","3559447266048","3545543175165","2649956144760","1398587965290","464904586800","73222472421"],["1","132","6237","166320","2962575","38394972","380750139","2983837824","18882098730","97907178600","420021796194","1499581950048","4467504559518","11102674053240","22941749956950","39153919926528","54601364897541","61213986943956","53845636663665","35797653183600","16914391129251","5062810950252","721764371007"],["1","144","7452","218592","4303530","61970832","686843388","6055435584","43428827079","257356012320","1273912260984","5306213054016","18682291794348","55715237067168","140709399735960","300180052770048","538213453990047","804526685549136","990759714611436","988015227867360","778061991945546","465778607423184","199206966397932","54226471004352","7060738412025"],["1","156","8775","280800","6054750","95907240","1174863690","11508868800","92250776475","615005176500","3450179040165","16423992786240","66722470694100","232146821113200","692887195669500","1773791220913920","3887097167705895","7263088133429700","11499889544597025","15290711859852000","16858009825486830","15137804741253480","10790377346554650","5874534358804800","2294739983908125","572767099983468","68630377364883"],["1","168","10206","353808","8292375","143292240","1922504220","20715963840","183530492145","1359485127000","8523971746290","45649038773520","210175782686415","835728556007520","2878146812781000","8596065147505920","22262465596861035","49917223898844120","96599072174615010","160552474528446000","227583132644072205","272480485342562640","271917509133177180","222057398762821440","144568618986211875","72168654597916968","25942282643925774","5978468428229808","663426981193869"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","40","1024","22528","458752","8912896","167772160","3087007744","55834574848","996432412672","17592186044416","307863255777280","5348024557502464","92323792361095168","1585267068834414592"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","-3"],["0","6"]]},"terms":[{"coeff":{"coeffs":[["1","9","27","27"],["0","6","36","54"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188843,"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-092f4bade7ac6bc8"]},{"id":188844,"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-092f4bade7ac6bc8"]},{"id":188845,"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-092f4bade7ac6bc8"]}],"region_exact":{},"incoming_edges":[{"id":"e-092f4bade7ac6bc8","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-7076c2900694d81e","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":"c7014183dd51bfd7c9be9086550758f7ceac4945b5c9f4a99469384b6fd84626","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-eb3f8d5fd252d0c9","role":"sequence","name":"basis_map_A013610 of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","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: 1,  12,   27;\n  2: 1,  24,  162,   432,     405;\n  3: 1,  36,  405,  2160,    6075,     8748,     5103;\n  4: 1,  48,  756,  6048,   28350,    81648,   142884,    139968,      59049;\n  5: 1,  60, 1215, 12960,   85050,   367416,  1071630,   2099520,    2657205,    1968300,      649539;\n  6: 1,  72, 1782, 23760,  200475,  1154736,  4715172,  13856832,   29229255,   43302600,    42869574,    25509168,      6908733;\n  7: 1,  84, 2457, 39312,  405405,  2918916, 15324309,  60046272,  177324147,  394053660,   650188539,   773778096,    628694703,    312487308,     71744535;\n  8: 1,  96, 3240, 60480,  737100,  6368544, 40864824, 200154240,  759960630, 2251735200,  5201508312,  9285337152,  12573894060,  12499492320,   8609344200,   3673320192,    731794257;\n  9: 1, 108, 4131, 88128, 1239300, 12492144, 94732092, 556792704, 2583866142, 9569874600, 28422527562, 67650313536, 128253719412, 191242232496, 219538277100, 187339329792, 111964521321, 41841412812, 7360989291;\n","recurrence_sentence":"The row polynomials satisfy (1+(6*n-3)*x)*P(n,x) = (1+(6*n+9)*x+(36*n+27)*x^2+(54*n+27)*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[((1 + (6*n + 9)*x + (36*n + 27)*x^2 + (54*n + 27)*x^3)*P[n - 1, x])/(1 + (6*n - 3)*x)]];\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] = ((1 + (6*n + 9)*x + (36*n + 27)*x^2 + (54*n + 27)*x^3)*v[i - 1])/(1 + (6*n - 3)*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-cdb0db4ad473ee28","steps":["basis A013610","\u03b8+1"],"edges":["e-4aebfd11077ddb65","e-092f4bade7ac6bc8"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A013610 \u2192 \u03b8+1","chain":null}
