{"id":"n-fbc65803a7c7ccb7","revision":0,"first_index":0,"fingerprint":"f6-23cc3a53c9ba374d","name":"theta_plus_one_squared of basis_map_A013610 of Triangle of coefficients in expansion of (1+5x)^n.","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":false,"rows":[["36","720","2025"],["216","6480","36450","54000"],["1296","51840","437400","1296000","1265625"],["7776","388800","4374000","19440000","37968750","27337500"],["46656","2799360","39366000","233280000","683437500","984150000","558140625"],["279936","19595520","330674400","2449440000","9568125000","20667150000","23441906250","10935000000"],["1679616","134369280","2645395200","23514624000","114817500000","330674400000","562605750000","524880000000","207594140625"],["10077696","906992640","20407334400","211631616000","1240029000000","4464104400000","10126903500000","14171760000000","11210083593750","3844335937500"],["60466176","6046617600","153055008000","1813985280000","12400290000000","53569252800000","151903552500000","283435200000000","336302507812500","230660156250000","69774697265625"],["362797056","39907676160","1122403392000","14965378560000","116917020000000","589261780800000","2005126893000000","4676680800000000","7398655171875000","7611785156250000","4605130019531250","1245564843750000"],["2176782336","261213880320","8081304422400","119723028480000","1052253180000000","6060978316800000","24061522716000000","67344203520000000","133175793093750000","182682843750000000","165784680703125000","89680668750000000","21927131103515625"],["13060694016","1697890222080","57303794995200","933839622144000","9119527560000000","59094538588800000","268114110264000000","875474645760000000","2077542372262500000","3562315453125000000","4310401698281250000","3497546081250000000","1710316226074218750","381454233398437500"],["78364164096","10970982973440","401126564966400","7131138932736000","76604031504000000","551549026828800000","2815198157772000000","10505695749120000000","29085593211675000000","59846899612500000000","90518435663906250000","97931290275000000000","71833281495117187500","32042155605468750000","6568408355712890625"],["470184984576","70527747686400","2777030065152000","53483541995520000","626760257760000000","4963941241459200000","28151981577720000000","118189077177600000000","373957627007250000000","897703494187500000000","1629331841950312500000","2203454031187500000000","2154998444853515625000","1441897002246093750000","591156752014160156250","112100835937500000000"],["2821109907456","451377585192960","19042491875328000","394955387043840000","5014082062080000000","43321669016371200000","270259023146112000000","1260683489894400000000","4487491524087000000000","12311362206000000000000","26069309471205000000000","42306317398800000000000","51719962676484375000000","46140704071875000000000","28375524096679687500000","10761680250000000000000","1898270014801025390625"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["2781","97146","3052161","89516826","2501220141","67412171106","1766867858721","45290172451986","1140018803999901","28265249737631466","691898142856755681","16752371740932261546","401782654499449718061","9556588121208364574226","225650314426128665359041"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["36","720","2025"]],"recurrence":{"denominator":{"coeffs":[["4","50","100"],["0","30","100"],["0","0","25"]]},"terms":[{"coeff":{"coeffs":[["24","540","2550","3375"],["0","180","1350","2250"],["0","0","150","375"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193478,"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-e6dead023820957c"]},{"id":193479,"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-e6dead023820957c"]},{"id":193480,"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-e6dead023820957c"]},{"id":193481,"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-e6dead023820957c"]}],"region_exact":{},"incoming_edges":[{"id":"e-e6dead023820957c","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-fbc65803a7c7ccb7","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":"d206e1b99be5d60d17e493bb622f6aed0d30df5dc84d93eeaf134d4cfc092c69","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-31eaa59be7a07b63","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (1+5x)^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":[{"id":"l-4d44e65ac2c484ed","kind":"fingerprint_related","node_a_id":"n-d99f7b722c4c442c","node_b_id":"n-fbc65803a7c7ccb7","fingerprint":"f6-23cc3a53c9ba374d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-23cc3a53c9ba374d"},"other_id":"n-d99f7b722c4c442c","other":{"id":"n-d99f7b722c4c442c","name":"theta_plus_one_squared of basis_map_A013611 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-23cc3a53c9ba374d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["8","80"],["64","1280","3600"]],"card_rows":[["1"],["8","80"],["64","1280","3600"],["512","15360","86400","128000"]],"row_sums_preview":["1","88","4944","230272","9646336","377223168","14053494784","505021038592"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-96f2019a633e1196","e-9e43149565a3ac9a"],"text":"A013622 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A013622 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","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:        36,         720,          2025;\n  1:       216,        6480,         36450,          54000;\n  2:      1296,       51840,        437400,        1296000,         1265625;\n  3:      7776,      388800,       4374000,       19440000,        37968750,        27337500;\n  4:     46656,     2799360,      39366000,      233280000,       683437500,       984150000,        558140625;\n  5:    279936,    19595520,     330674400,     2449440000,      9568125000,     20667150000,      23441906250,      10935000000;\n  6:   1679616,   134369280,    2645395200,    23514624000,    114817500000,    330674400000,     562605750000,     524880000000,     207594140625;\n  7:  10077696,   906992640,   20407334400,   211631616000,   1240029000000,   4464104400000,   10126903500000,   14171760000000,   11210083593750,    3844335937500;\n  8:  60466176,  6046617600,  153055008000,  1813985280000,  12400290000000,  53569252800000,  151903552500000,  283435200000000,  336302507812500,  230660156250000,   69774697265625;\n  9: 362797056, 39907676160, 1122403392000, 14965378560000, 116917020000000, 589261780800000, 2005126893000000, 4676680800000000, 7398655171875000, 7611785156250000, 4605130019531250, 1245564843750000;\n","recurrence_sentence":"The row polynomials satisfy (4+(30*n+50)*x+(25*n^2+100*n+100)*x^2)*P(n,x) = (24+(180*n+540)*x+(150*n^2+1350*n+2550)*x^2+(375*n^2+2250*n+3375)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 720*x + 2025*x^2.","mathematica":"P[0, x_] := 36 + 720*x + 2025*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((24 + (180*n + 540)*x + (150*n^2 + 1350*n + 2550)*x^2 + (375*n^2 + 2250*n + 3375)*x^3)*P[n - 1, x])/(4 + (30*n + 50)*x + (25*n^2 + 100*n + 100)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 36 + 720*x + 2025*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((24 + (180*n + 540)*x + (150*n^2 + 1350*n + 2550)*x^2 + (375*n^2 + 2250*n + 3375)*x^3)*v[i - 1])/(4 + (30*n + 50)*x + (25*n^2 + 100*n + 100)*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-bbfc5a54ff13a785","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-46de8f4df4fd1cb3","e-e6dead023820957c"],"text":"A013612 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A013612 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}
