{"id":"n-3e7c547c780b8c12","revision":0,"first_index":0,"fingerprint":"f6-235bca8d2225d93d","name":"one_plus_derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^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":[["799","960","225"],["17918","35955","21600","3375"],["378301","1075080","1078650","432000","50625"],["7683932","28372575","40315500","26966250","8100000","759375"],["151924699","691553880","1276765875","1209465000","606740625","145800000","11390625"],["2944783418","15952093395","36306578700","44686805625","31748456250","12741553125","2551500000","170859375"],["56216398201","353374010160","957125603700","1452263148000","1340604168750","761962950000","254831062500","43740000000","2562890625"],["1060315131032","7589213757135","23852745685800","43070652166500","49013881245000","36196312556250","17144166375000","4914599062500","738112500000","38443359375"],["19804175374999","159047269654800","569191031785125","1192637284290000","1615149456243750","1470416437350000","904907813906250","367374993750000","92148732421875","12301875000000","576650390625"],["366910889881718","3267688936874835","13121399746521000","31305506748181875","49196287976962500","53299932056043750","40436452027125000","21329969899218750","7577109246093750","1689393427734375","202980937500000","8649755859375"],["6751563572603701","66043960178709240","294092004318735150","787283984791260000","1408747803668184375","1771066367170650000","1598997961681312500","1039794480697500000","479924322732421875","151542184921875000","30409081699218750","3321506250000000","129746337890625"],["123515914292709332","1316554896657721695","6439286117424150900","19115980280717784750","38380094258573925000","54941164343059190625","57559656933046125000","44543514646836562500","25344990467001562500","10398360325869140625","2955072605976562500","539070084667968750","53974476562500000","1946195068359375"],["2248339213469647699","25938342001468959720","138238264149060777975","450750028219690563000","1003588964737683699375","1611963958860104850000","1922940752007071671875","1726789707991383750000","1169267259479459765625","591383110896703125000","218365566843251953125","56415022477734375000","9433726481689453125","871895390625000000","29192926025390625"],["40747434027375024818","505876323030670732275","2918063475165257968500","10367869811179558348125","25354689087357594168750","45161503413195766471875","60448648457253931875000","61808809885941589453125","48565960537257667968750","29231681486986494140625","13306119995175820312500","4466568412702880859375","1057781671457519531250","163276035260009765625","14012604492187500000","437893890380859375"],["735642724238022658801","9779384166570005956320","60705158763680487873000","233445078013220637480000","622072188670773500887500","1217025076193164520100000","1806460136527830658875000","2072525089962991950000000","1854264296578247683593750","1295092280993537812500000","701560355687675859375000","290315345349290625000000","89331368254057617187500","19528277011523437500000","2799017747314453125000","224201671875000000000","6568408355712890625"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1984","78848","3014656","112197632","4093640704","147102629888","5222680231936","183618441838592","6403555720167424","221802281647996928","7638104968020361216","261713181545754263552","8928224131675422982144","303412046524374704979968","10275869466724347985002496"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["799","960","225"]],"recurrence":{"denominator":{"coeffs":[["32","15"],["15","0"]]},"terms":[{"coeff":{"coeffs":[["799","960","225"],["255","225","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194386,"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-74f60d40993ea0dc"]},{"id":194387,"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-74f60d40993ea0dc"]},{"id":194388,"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-74f60d40993ea0dc"]},{"id":194389,"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-74f60d40993ea0dc"]}],"region_exact":{},"incoming_edges":[{"id":"e-74f60d40993ea0dc","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-3e7c547c780b8c12","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":"ec986b88eb26f8d434a0b5edd7e313e90824cf5cb837c748135d5cc4dba67e64","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-8a857e84342335b8","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^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":[],"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:             799,              960,               225;\n  1:           17918,            35955,             21600,              3375;\n  2:          378301,          1075080,           1078650,            432000,             50625;\n  3:         7683932,         28372575,          40315500,          26966250,           8100000,            759375;\n  4:       151924699,        691553880,        1276765875,        1209465000,         606740625,         145800000,          11390625;\n  5:      2944783418,      15952093395,       36306578700,       44686805625,       31748456250,       12741553125,        2551500000,         170859375;\n  6:     56216398201,     353374010160,      957125603700,     1452263148000,     1340604168750,      761962950000,      254831062500,       43740000000,       2562890625;\n  7:   1060315131032,    7589213757135,    23852745685800,    43070652166500,    49013881245000,    36196312556250,    17144166375000,     4914599062500,     738112500000,      38443359375;\n  8:  19804175374999,  159047269654800,   569191031785125,  1192637284290000,  1615149456243750,  1470416437350000,   904907813906250,   367374993750000,   92148732421875,   12301875000000,    576650390625;\n  9: 366910889881718, 3267688936874835, 13121399746521000, 31305506748181875, 49196287976962500, 53299932056043750, 40436452027125000, 21329969899218750, 7577109246093750, 1689393427734375, 202980937500000, 8649755859375;\n","recurrence_sentence":"The row polynomials satisfy (15*n+32+15*x)*P(n,x) = (255*n+799+(225*n+960)*x+225*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 799 + 960*x + 225*x^2.","mathematica":"P[0, x_] := 799 + 960*x + 225*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((255*n + 799 + (225*n + 960)*x + 225*x^2)*P[n - 1, x])/(15*n + 32 + 15*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 799 + 960*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((255*n + 799 + (225*n + 960)*x + 225*x^2)*v[i - 1])/(15*n + 32 + 15*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-a5302200d285579d","steps":["basis A013610","D+1"],"edges":["e-633275fbd6b7b077","e-74f60d40993ea0dc"],"text":"A038331 \u2192 basis A013610 \u2192 D+1"},"headline":"A038331 \u2192 basis A013610 \u2192 D+1","chain":null}