{"id":"n-af83e2dbf104af1b","revision":0,"first_index":0,"fingerprint":"f6-e2c46986e009fe93","name":"shift_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(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":[["256","96","9"],["4096","2304","432","27"],["65536","49152","13824","1728","81"],["1048576","983040","368640","69120","6480","243"],["16777216","18874368","8847360","2211840","311040","23328","729"],["268435456","352321536","198180864","61931520","11612160","1306368","81648","2187"],["4294967296","6442450944","4227858432","1585446912","371589120","55738368","5225472","279936","6561"],["68719476736","115964116992","86973087744","38050725888","10701766656","2006581248","250822656","20155392","944784","19683"],["1099511627776","2061584302080","1739461754880","869730877440","285380444160","64210599936","10032906240","1074954240","75582720","3149280","59049"],["17592186044416","36283883716608","34016140984320","19134079303680","7175279738880","1883510931456","353158299648","47297986560","4434186240","277136640","10392624","177147"],["281474976710656","633318697598976","653109906898944","408193691811840","172206713733120","51662014119936","11301065588736","1816242683904","212840939520","17736744960","997691904","34012224","531441"],["4503599627370496","10977524091715584","12349714603180032","8490428789686272","3979888495165440","1343212367118336","335803091779584","62963079708672","8854183084032","922310737920","69173305344","3537271296","110539728","1594323"],["72057594037927936","189151184349560832","230528005926027264","172896004444520448","89149502291705856","33431063359389696","9402486569828352","2014818550677504","330556168470528","41319521058816","3873705099264","264116256768","12380449536","357128352","4782969"],["1152921504606846976","3242591731706757120","4255901647865118720","3457920088890408960","1945080050000855040","802345520625352704","250732975195422720","60444556520325120","11333354347560960","1652780842352640","185937844764672","15846975406080","990435962880","42855402240","1147912560","14348907"],["18446744073709551616","55340232221128654848","77822201560962170880","68094426365841899520","41495041066684907520","18672768480008208384","6418764165002821632","1719311829911470080","362667339121950720","60444556520325120","7933348043292672","811365140791296","63387901624320","3656994324480","146932807680","3673320192","43046721"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["361","6859","130321","2476099","47045881","893871739","16983563041","322687697779","6131066257801","116490258898219","2213314919066161","42052983462257059","799006685782884121","15181127029874798299","288441413567621167681"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["256","96","9"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["16","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192577,"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-303aeaf50e17fb7f"]},{"id":192578,"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-303aeaf50e17fb7f","e-d20c30b308461949"]},{"id":192579,"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-303aeaf50e17fb7f","e-d20c30b308461949"]},{"id":192580,"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-303aeaf50e17fb7f"]},{"id":192581,"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-d20c30b308461949"]}],"region_exact":{},"incoming_edges":[{"id":"e-d20c30b308461949","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-af83e2dbf104af1b","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":"56d8b7dd7913d3ea65b16a751050cbaa28b28f962c1ef987e56d562f4185e05e","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-4ba4cfdaaf5fda7d","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(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]"}]},{"id":"e-303aeaf50e17fb7f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-af83e2dbf104af1b","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":"8f01547d95c725d1e5b3aeacd850fc9baf781c75940a99608c229b938151a370","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-4ba4cfdaaf5fda7d","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(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-2c6ff9d7e8df11fc","kind":"fingerprint_related","node_a_id":"n-d48c40c60a8e7740","node_b_id":"n-af83e2dbf104af1b","fingerprint":"f6-e2c46986e009fe93","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e2c46986e009fe93"},"other_id":"n-d48c40c60a8e7740","other":{"id":"n-d48c40c60a8e7740","name":"derivative of basis_map_A013610 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-e2c46986e009fe93","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["96","18"]],"card_rows":[["3"],["96","18"],["2304","864","81"],["49152","27648","5184","324"]],"row_sums_preview":["0","3","114","3249","82308","1954815","44569782","987963501"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013610","D"],"edges":["e-d2b5f968b40c2621","e-345aa87df8a9647d"],"text":"A027467 \u2192 basis A013610 \u2192 D"},"headline":"A027467 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-d68272449fa70b1c","kind":"fingerprint_related","node_a_id":"n-16c98a2b049669c5","node_b_id":"n-af83e2dbf104af1b","fingerprint":"f6-e2c46986e009fe93","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e2c46986e009fe93"},"other_id":"n-16c98a2b049669c5","other":{"id":"n-16c98a2b049669c5","name":"basis_map_A013610 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-e2c46986e009fe93","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","3"],["256","96","9"]],"card_rows":[["1"],["16","3"],["256","96","9"],["4096","2304","432","27"]],"row_sums_preview":["1","19","361","6859","130321","2476099","47045881","893871739"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013610"],"edges":["e-d2b5f968b40c2621"],"text":"A027467 \u2192 basis A013610"},"headline":"A027467 \u2192 basis A013610","chain":null}},{"id":"l-f398793a62f207bd","kind":"fingerprint_related","node_a_id":"n-11267945ca8c7a9f","node_b_id":"n-af83e2dbf104af1b","fingerprint":"f6-e2c46986e009fe93","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e2c46986e009fe93"},"other_id":"n-11267945ca8c7a9f","other":{"id":"n-11267945ca8c7a9f","name":"reversal of basis_map_A013610 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-e2c46986e009fe93","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","16"],["9","96","256"]],"card_rows":[["1"],["3","16"],["9","96","256"],["27","432","2304","4096"]],"row_sums_preview":["1","19","361","6859","130321","2476099","47045881","893871739"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013610","reverse"],"edges":["e-d2b5f968b40c2621","e-31db37ef73cb4075"],"text":"A027467 \u2192 basis A013610 \u2192 reverse"},"headline":"A027467 \u2192 basis A013610 \u2192 reverse","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:            256,             96,              9;\n  1:           4096,           2304,            432,             27;\n  2:          65536,          49152,          13824,           1728,            81;\n  3:        1048576,         983040,         368640,          69120,          6480,           243;\n  4:       16777216,       18874368,        8847360,        2211840,        311040,         23328,          729;\n  5:      268435456,      352321536,      198180864,       61931520,      11612160,       1306368,        81648,        2187;\n  6:     4294967296,     6442450944,     4227858432,     1585446912,     371589120,      55738368,      5225472,      279936,       6561;\n  7:    68719476736,   115964116992,    86973087744,    38050725888,   10701766656,    2006581248,    250822656,    20155392,     944784,     19683;\n  8:  1099511627776,  2061584302080,  1739461754880,   869730877440,  285380444160,   64210599936,  10032906240,  1074954240,   75582720,   3149280,    59049;\n  9: 17592186044416, 36283883716608, 34016140984320, 19134079303680, 7175279738880, 1883510931456, 353158299648, 47297986560, 4434186240, 277136640, 10392624, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (16+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 96*x + 9*x^2.","mathematica":"P[0, x_] := 256 + 96*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[(16 + 3*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] = 256 + 96*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 3*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-4397e742863fdbe2","steps":["x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-6056f3721ff524be","e-303aeaf50e17fb7f"],"text":"A038305 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"A038305 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","chain":null}
