{"id":"n-7685734ea2b0f312","revision":0,"first_index":0,"fingerprint":"f6-700950e0f523054a","name":"basis_map_A013609 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^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"],["17","10"],["289","340","100"],["4913","8670","5100","1000"],["83521","196520","173400","68000","10000"],["1419857","4176050","4913000","2890000","850000","100000"],["24137569","85191420","125281500","98260000","43350000","10200000","1000000"],["410338673","1689629830","2981699700","2923235000","1719550000","606900000","119000000","10000000"],["6975757441","32827093840","67585193200","79511992000","58464700000","27512800000","8092000000","1360000000","100000000"],["118587876497","627818169690","1477219222800","2027555796000","1789019820000","1052364600000","412692000000","104040000000","15300000000","1000000000"],["2015993900449","11858787649700","31390908484500","49240640760000","50688894900000","35780396400000","17539410000000","5895600000000","1300500000000","170000000000","10000000000"],["34271896307633","221759329049390","652233320733500","1150999977765000","1354117620900000","1115155687800000","655973934000000","275619300000000","81064500000000","15895000000000","1870000000000","100000000000"],["582622237229761","4112627556915960","13305559742963400","26089332829340000","34529999332950000","32498822901600000","22303113756000000","11245267440000000","4134289500000000","1080860000000000","190740000000000","20400000000000","1000000000000"],["9904578032905937","75740890839868930","267320791199537400","576574255528414000","847903316953550000","897779982656700000","704141162868000000","414200684040000000","182735595900000000","59717515000000000","14051180000000000","2254200000000000","221000000000000","10000000000000"],["168377826559400929","1386640924606831180","5301862358790825100","12474970255978412000","20180098943494490000","23741292874699400000","20948199595323000000","14082823257360000000","7248511970700000000","2842553714000000000","836045210000000000","178833200000000000","26299000000000000","2380000000000000","100000000000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","27","729","19683","531441","14348907","387420489","10460353203","282429536481","7625597484987","205891132094649","5559060566555523","150094635296999121","4052555153018976267","109418989131512359209"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["17","10"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":200731,"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-3fbf6fb5e0487184"]},{"id":200732,"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-3fbf6fb5e0487184"]},{"id":200733,"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-3fbf6fb5e0487184"]}],"region_exact":{},"incoming_edges":[{"id":"e-3fbf6fb5e0487184","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-7685734ea2b0f312","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"3fd6fcadebdd41eb47e665366cb139f608cb298fc8d9e625e42218a61642a396","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-7d034d48b0ca9aab","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(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":[{"id":"l-46990bd8024804f3","kind":"fingerprint_related","node_a_id":"n-45bc9eeddd99bec8","node_b_id":"n-7685734ea2b0f312","fingerprint":"f6-700950e0f523054a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-700950e0f523054a"},"other_id":"n-45bc9eeddd99bec8","other":{"id":"n-45bc9eeddd99bec8","name":"reversal of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.","first_index":0,"fingerprint":"f6-700950e0f523054a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["100","340","289"],["1000","5100","8670","4913"],["10000","68000","173400","196520","83521"]],"card_rows":[["100","340","289"],["1000","5100","8670","4913"],["10000","68000","173400","196520","83521"],["100000","850000","2890000","4913000","4176050","1419857"]],"row_sums_preview":["729","19683","531441","14348907","387420489","10460353203","282429536481","7625597484987"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a5302200d285579d","steps":["basis A013609","reverse"],"edges":["e-2f341f062a4ee4be","e-518b16e8aab2c81d"],"text":"A038331 \u2192 basis A013609 \u2192 reverse"},"headline":"A038331 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-8d03a57dd3397dd6","kind":"fingerprint_related","node_a_id":"n-880addc4a45abb21","node_b_id":"n-7685734ea2b0f312","fingerprint":"f6-700950e0f523054a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-700950e0f523054a"},"other_id":"n-880addc4a45abb21","other":{"id":"n-880addc4a45abb21","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.","first_index":0,"fingerprint":"f6-700950e0f523054a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["340","200"],["8670","10200","3000"],["196520","346800","204000","40000"]],"card_rows":[["340","200"],["8670","10200","3000"],["196520","346800","204000","40000"],["4176050","9826000","8670000","3400000","500000"]],"row_sums_preview":["540","21870","787320","26572050","860934420","27119434230","836828256240","25418658283290"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a5302200d285579d","steps":["basis A013609","D"],"edges":["e-2f341f062a4ee4be","e-5bc9576b6968b048"],"text":"A038331 \u2192 basis A013609 \u2192 D"},"headline":"A038331 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-d4c0dc6bbc1b344a","kind":"fingerprint_related","node_a_id":"n-688fdf685007c036","node_b_id":"n-7685734ea2b0f312","fingerprint":"f6-700950e0f523054a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-700950e0f523054a"},"other_id":"n-688fdf685007c036","other":{"id":"n-688fdf685007c036","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.","first_index":0,"fingerprint":"f6-700950e0f523054a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["289","340","100"],["4913","8670","5100","1000"],["83521","196520","173400","68000","10000"]],"card_rows":[["289","340","100"],["4913","8670","5100","1000"],["83521","196520","173400","68000","10000"],["1419857","4176050","4913000","2890000","850000","100000"]],"row_sums_preview":["729","19683","531441","14348907","387420489","10460353203","282429536481","7625597484987"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a5302200d285579d","steps":["basis A013609"],"edges":["e-2f341f062a4ee4be"],"text":"A038331 \u2192 basis A013609"},"headline":"A038331 \u2192 basis A013609","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:           17,           10;\n  2:          289,          340,           100;\n  3:         4913,         8670,          5100,          1000;\n  4:        83521,       196520,        173400,         68000,         10000;\n  5:      1419857,      4176050,       4913000,       2890000,        850000,        100000;\n  6:     24137569,     85191420,     125281500,      98260000,      43350000,      10200000,      1000000;\n  7:    410338673,   1689629830,    2981699700,    2923235000,    1719550000,     606900000,    119000000,     10000000;\n  8:   6975757441,  32827093840,   67585193200,   79511992000,   58464700000,   27512800000,   8092000000,   1360000000,   100000000;\n  9: 118587876497, 627818169690, 1477219222800, 2027555796000, 1789019820000, 1052364600000, 412692000000, 104040000000, 15300000000, 1000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (17+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(17 + 10*x)*P[n - 1, 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] = (17 + 10*x)*v[i - 1]); 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-a6a633b989da1879","steps":["basis A007318","basis A013609"],"edges":["e-7ab50ded9e353f77","e-3fbf6fb5e0487184"],"text":"A038271 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A038271 \u2192 basis A007318 \u2192 basis A013609","chain":null}