{"id":"n-5de516ea0ecc91dc","revision":0,"first_index":0,"fingerprint":"f6-bb24aeb74e5c1fb8","name":"one_plus_derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^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":[["187","84","9"],["2420","1683","378","27"],["30613","29040","10098","1512","81"],["380666","459195","217800","50490","5670","243"],["4670479","6851988","4132755","1306800","227205","20412","729"],["56689952","98080059","71945874","28929285","6860700","954261","71442","2187"],["682050985","1360558848","1176960708","575566992","173575710","32931360","3817044","244944","6561"],["8145637478","18415376595","18367544448","10592646372","3885077196","937308834","148191120","14722884","826686","19683"],["96675855331","244369124340","276230648925","183675444480","79444847790","23310463176","4686544170","635104800","55210815","2755620","59049"],["1141246682444","3190303225923","4032090551610","3038537138175","1515322416960","524335995414","128207547468","22093708230","2619807300","202439655","9093546","177147"],["13409648518717","41084880567984","57425458066614","48385086619320","27346834243575","10910321402112","3146015972484","659353101264","99421687035","10479229200","728782758","29760696","531441"],["156921418836050","522976292229963","801155171075688","746530954865982","471754594538370","213305307099885","70917089113728","17527803275268","3214346368662","430827310485","40868993880","2583866142","96722262","1594323"],["1829703743628343","6590699591114100","10982502136829223","11216172395059632","7838575026092811","3962738594122308","1493137149699195","425502534682368","92020967195157","15000283053756","1809474704037","156045249360","9043531497","312487308","4782969"],["21265990680661496","82336668463275435","148290740800067250","164737532052438345","126181939444420860","70547175234835299","29720539455917310","9598738819494825","2393451757588320","460104835975785","67501273741902","7402396516515","585169685100","31304532105","1004423490","14348907"],["246457641995523409","1020767552671751808","1976080043118610440","2372651852801076000","1976850384629260140","1211346618666440256","564377401878682392","203797984840575840","57592432916968950","12765076040471040","2208503212683768","294551012691936","29609586066060","2160626529600","107329824360","3214155168","43046721"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["280","4508","71344","1114064","17210368","263533760","4005713152","60507351296","909086058496","13594968783872","202478258483200","3004777355890688","44448027302232064","655608402707922944","9645221924584357888"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["187","84","9"]],"recurrence":{"denominator":{"coeffs":[["14","3"],["3","0"]]},"terms":[{"coeff":{"coeffs":[["187","84","9"],["33","9","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191991,"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-10e9f5827c0e3ee5"]},{"id":191992,"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-10e9f5827c0e3ee5"]},{"id":191993,"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-10e9f5827c0e3ee5"]},{"id":191994,"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-10e9f5827c0e3ee5"]}],"region_exact":{},"incoming_edges":[{"id":"e-10e9f5827c0e3ee5","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-5de516ea0ecc91dc","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"053d192f1a27a0ab25660291344dc78bbf298881057cc315d457e947f9759cbb","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-809ca598570cd420","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^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-789c5834682ae6c3","kind":"fingerprint_related","node_a_id":"n-83fdacfbb15a5bb9","node_b_id":"n-5de516ea0ecc91dc","fingerprint":"f6-bb24aeb74e5c1fb8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-bb24aeb74e5c1fb8"},"other_id":"n-83fdacfbb15a5bb9","other":{"id":"n-83fdacfbb15a5bb9","name":"one_plus_derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","first_index":0,"fingerprint":"f6-bb24aeb74e5c1fb8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["14","3"],["187","84","9"]],"card_rows":[["1"],["14","3"],["187","84","9"],["2420","1683","378","27"]],"row_sums_preview":["1","17","280","4508","71344","1114064","17210368","263533760"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["basis A013610","D+1"],"edges":["e-254f1b828fe6620e","e-795c5f3a4743ff4b"],"text":"A038303 \u2192 basis A013610 \u2192 D+1"},"headline":"A038303 \u2192 basis A013610 \u2192 D+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:           187,            84,             9;\n  1:          2420,          1683,           378,            27;\n  2:         30613,         29040,         10098,          1512,            81;\n  3:        380666,        459195,        217800,         50490,          5670,          243;\n  4:       4670479,       6851988,       4132755,       1306800,        227205,        20412,          729;\n  5:      56689952,      98080059,      71945874,      28929285,       6860700,       954261,        71442,        2187;\n  6:     682050985,    1360558848,    1176960708,     575566992,     173575710,     32931360,      3817044,      244944,       6561;\n  7:    8145637478,   18415376595,   18367544448,   10592646372,    3885077196,    937308834,    148191120,    14722884,     826686,     19683;\n  8:   96675855331,  244369124340,  276230648925,  183675444480,   79444847790,  23310463176,   4686544170,   635104800,   55210815,   2755620,   59049;\n  9: 1141246682444, 3190303225923, 4032090551610, 3038537138175, 1515322416960, 524335995414, 128207547468, 22093708230, 2619807300, 202439655, 9093546, 177147;\n","recurrence_sentence":"The row polynomials satisfy (3*n+14+3*x)*P(n,x) = (33*n+187+(9*n+84)*x+9*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 187 + 84*x + 9*x^2.","mathematica":"P[0, x_] := 187 + 84*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((33*n + 187 + (9*n + 84)*x + 9*x^2)*P[n - 1, x])/(3*n + 14 + 3*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 187 + 84*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((33*n + 187 + (9*n + 84)*x + 9*x^2)*v[i - 1])/(3*n + 14 + 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-108b34f10b7468f4","steps":["x \u21a6 x+1","D+1"],"edges":["e-3a9169eda66e0655","e-10e9f5827c0e3ee5"],"text":"A038281 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A038281 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}