{"id":"n-d5d2f240e694c36e","revision":0,"first_index":0,"fingerprint":"f6-bbe11d3a6446c7ec","name":"shift_x_plus_one 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":[["1024","960","225"],["32768","46080","21600","3375"],["1048576","1966080","1382400","432000","50625"],["33554432","78643200","73728000","34560000","8100000","759375"],["1073741824","3019898880","3538944000","2211840000","777600000","145800000","11390625"],["34359738368","112742891520","158544691200","123863040000","58060800000","16329600000","2551500000","170859375"],["1099511627776","4123168604160","6764573491200","6341787648000","3715891200000","1393459200000","326592000000","43740000000","2562890625"],["35184372088832","148434069749760","278313880780800","304405807104000","214035333120000","100329062400000","31352832000000","6298560000000","738112500000","38443359375"],["1125899906842624","5277655813324800","11132555231232000","13915694039040000","11415217766400000","6421059993600000","2508226560000000","671846400000000","118098000000000","12301875000000","576650390625"],["36028797018963968","185773484629032960","435406604599296000","612290537717760000","574022379110400000","376702186291200000","176579149824000000","59122483200000000","13856832000000000","2165130000000000","202980937500000","8649755859375"],["1152921504606846976","6485183463413514240","16719613616612966400","26124396275957760000","27553074197299200000","20664805647974400000","11301065588736000000","4540606709760000000","1330255872000000000","277136640000000000","38972340000000000","3321506250000000","129746337890625"],["36893488147419103232","224819693398335160320","632305387682817638400","1086774885079842816000","1273564318452940800000","1074569893694668800000","671606183559168000000","314815398543360000000","110677288550400000000","28822210560000000000","5404164480000000000","690873300000000000","53974476562500000","1946195068359375"],["1180591620717411303424","7747632510958011678720","23606067806825191833600","44261377137797234688000","57055681466691747840000","53489701375023513600000","37609946279313408000000","20148185506775040000000","8263904211763200000000","2582470066176000000000","605266421760000000000","103170412800000000000","12090282750000000000","871895390625000000","29192926025390625"],["37778931862957161709568","265633114661417543270400","871608657482776313856000","1770455085511889387520000","2489702464001094451200000","2567505666001128652800000","2005863801563381760000000","1208891130406502400000000","566667717378048000000000","206597605294080000000000","58105576488960000000000","12380449536000000000000","1934445240000000000000","209254893750000000000","14012604492187500000","437893890380859375"],["1208925819614629174706176","9066943647109718810296320","31875973759370105192448000","69728692598622105108480000","106227305130713363251200000","119505718272052533657600000","102700226640045146112000000","68772473196458803200000000","36266733912195072000000000","15111139130081280000000000","4958342527057920000000000","1267758032486400000000000","247608990720000000000000","35712835200000000000000","3587226750000000000000","224201671875000000000","6568408355712890625"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["2209","103823","4879681","229345007","10779215329","506623120463","23811286661761","1119130473102767","52599132235830049","2472159215084012303","116191483108948578241","5460999706120583177327","256666986187667409334369","12063348350820368238715343","566977372488557307219621121"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1024","960","225"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["32","15"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194394,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-107659d6e66cdf16"]},{"id":194395,"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-107659d6e66cdf16","e-59a252893bc6e87f"]},{"id":194396,"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-107659d6e66cdf16","e-59a252893bc6e87f"]},{"id":194397,"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-107659d6e66cdf16"]},{"id":194398,"position":4,"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-59a252893bc6e87f"]}],"region_exact":{},"incoming_edges":[{"id":"e-59a252893bc6e87f","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-d5d2f240e694c36e","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":"2979910b38488c4ae16b77e00c87aa42873bc35646ab7af1f37d9741c4000928","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-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]"}]},{"id":"e-107659d6e66cdf16","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-d5d2f240e694c36e","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"742074c4e559e5f70733d7f3625c8f77c588217b05fbdf1dd1951c3541da0f95","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"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:              1024,                960,                225;\n  1:             32768,              46080,              21600,               3375;\n  2:           1048576,            1966080,            1382400,             432000,              50625;\n  3:          33554432,           78643200,           73728000,           34560000,            8100000,             759375;\n  4:        1073741824,         3019898880,         3538944000,         2211840000,          777600000,          145800000,           11390625;\n  5:       34359738368,       112742891520,       158544691200,       123863040000,        58060800000,        16329600000,         2551500000,         170859375;\n  6:     1099511627776,      4123168604160,      6764573491200,      6341787648000,      3715891200000,      1393459200000,       326592000000,       43740000000,        2562890625;\n  7:    35184372088832,    148434069749760,    278313880780800,    304405807104000,    214035333120000,    100329062400000,     31352832000000,     6298560000000,      738112500000,      38443359375;\n  8:  1125899906842624,   5277655813324800,  11132555231232000,  13915694039040000,  11415217766400000,   6421059993600000,   2508226560000000,   671846400000000,   118098000000000,   12301875000000,    576650390625;\n  9: 36028797018963968, 185773484629032960, 435406604599296000, 612290537717760000, 574022379110400000, 376702186291200000, 176579149824000000, 59122483200000000, 13856832000000000, 2165130000000000, 202980937500000, 8649755859375;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (32+15*x)*P(n-1,x) for n >= 1, with P(0,x) = 1024 + 960*x + 225*x^2.","mathematica":"P[0, x_] := 1024 + 960*x + 225*x^2;\nP[n_, x_] := P[n, x] = Expand[(32 + 15*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] = 1024 + 960*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = (32 + 15*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-a5302200d285579d","steps":["basis A013610","x \u21a6 x+1"],"edges":["e-633275fbd6b7b077","e-107659d6e66cdf16"],"text":"A038331 \u2192 basis A013610 \u2192 x \u21a6 x+1"},"headline":"A038331 \u2192 basis A013610 \u2192 x \u21a6 x+1","chain":null}
