{"id":"n-cd59b83bc19527ec","revision":0,"first_index":0,"fingerprint":"f6-7c3b314392150c36","name":"one_plus_derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","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":true,"rows":[["1"],["9","2"],["77","36","4"],["637","462","108","8"],["5145","5096","1848","288","16"],["40817","51450","25480","6160","720","32"],["319333","489804","308700","101920","18480","1728","64"],["2470629","4470662","3428628","1440600","356720","51744","4032","128"],["18941489","39530064","35765296","18286016","5762400","1141504","137984","9216","256"],["144120025","340946802","355770576","214591776","82287072","20744640","3424512","354816","20736","512"],["1089547389","2882400500","3409468020","2371803840","1072958880","329148288","69148800","9784320","887040","46080","1024"],["8191782221","23970042558","31706405500","25002765480","13044921120","4721019072","1206877056","217324800","26906880","2168320","101376","2048"],["61297129033","196602773304","287640510696","253651244000","150016592880","62615621376","18884076288","4137864192","651974400","71751680","5203968","221184","4096"],["456762477633","1593725354858","2555836052952","2492884426032","1648733086000","780086282976","271334359296","70140854784","13448058624","1883481600","186554368","12300288","479232","8192"],["3391115364245","12789349373724","22312154968012","23854469827552","17450190982224","9232905281600","3640402653888","1085337437184","245492991744","41838404608","5273748480","474865664","28700672","1032192","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","117","1215","12393","124659","1240029","12223143","119574225","1162261467","11235194181","108090316431","1035574967097","9885033776835","94049035648173"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["5","2"],["2","0"]]},"terms":[{"coeff":{"coeffs":[["49","28","4"],["14","4","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184304,"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-63647191d16071b7"]},{"id":184305,"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-63647191d16071b7"]},{"id":184306,"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-63647191d16071b7"]},{"id":184307,"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-63647191d16071b7"]}],"region_exact":{},"incoming_edges":[{"id":"e-63647191d16071b7","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-cd59b83bc19527ec","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":"ce64fff508d2f9378a67bccf38bf938f3bfbcbcf6b5e166289274fffca781fda","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-6a4f4798749d2440","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-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-bfeaa60e68e06292","kind":"fingerprint_related","node_a_id":"n-cd59b83bc19527ec","node_b_id":"n-b2c5c5625a1a0c85","fingerprint":"f6-7c3b314392150c36","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7c3b314392150c36"},"other_id":"n-b2c5c5625a1a0c85","other":{"id":"n-b2c5c5625a1a0c85","name":"one_plus_derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-7c3b314392150c36","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["308","144","16"],["5096","3696","864","64"],["82320","81536","29568","4608","256"]],"card_rows":[["308","144","16"],["5096","3696","864","64"],["82320","81536","29568","4608","256"],["1306144","1646400","815360","197120","23040","1024"]],"row_sums_preview":["468","9720","198288","3989088","79361856","1564562304","30611001600","595077871104"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["basis A013609","D+1"],"edges":["e-ccf7b0982517b62a","e-46a05f75ad982780"],"text":"A038328 \u2192 basis A013609 \u2192 D+1"},"headline":"A038328 \u2192 basis A013609 \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:         1;\n  1:         9,         2;\n  2:        77,        36,         4;\n  3:       637,       462,       108,         8;\n  4:      5145,      5096,      1848,       288,       16;\n  5:     40817,     51450,     25480,      6160,      720,       32;\n  6:    319333,    489804,    308700,    101920,    18480,     1728,      64;\n  7:   2470629,   4470662,   3428628,   1440600,   356720,    51744,    4032,    128;\n  8:  18941489,  39530064,  35765296,  18286016,  5762400,  1141504,  137984,   9216,   256;\n  9: 144120025, 340946802, 355770576, 214591776, 82287072, 20744640, 3424512, 354816, 20736, 512;\n","recurrence_sentence":"The row polynomials satisfy (2*n+5+2*x)*P(n,x) = (14*n+49+(4*n+28)*x+4*x^2)*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[((14*n + 49 + (4*n + 28)*x + 4*x^2)*P[n - 1, x])/(2*n + 5 + 2*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] = ((14*n + 49 + (4*n + 28)*x + 4*x^2)*v[i - 1])/(2*n + 5 + 2*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-22693abd431a71f2","steps":["basis A013609","D+1"],"edges":["e-4c464713c83c5f6b","e-63647191d16071b7"],"text":"A038255 \u2192 basis A013609 \u2192 D+1"},"headline":"A038255 \u2192 basis A013609 \u2192 D+1","chain":null}
