{"id":"n-cdff05d2922e65c0","revision":0,"first_index":0,"fingerprint":"f6-f3c17ca3ebbc301e","name":"shift_x_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^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"],["48","27"],["2304","2592","729"],["110592","186624","104976","19683"],["5308416","11943936","10077696","3779136","531441"],["254803968","716636160","806215680","453496320","127545840","14348907"],["12230590464","41278242816","58047528960","43535646720","18366600960","4132485216","387420489"],["587068342272","2311581597696","3900793946112","3656994324480","2057059307520","694257516288","130173284304","10460353203"],["28179280429056","126806761930752","249650812551168","280857164120064","197477693521920","88864962084864","24993270586368","4016775629952","282429536481"],["1352605460594688","6847565144260608","15407021574586368","20221715816644608","17062072720293888","9597415905165312","3599030964436992","867623536069632","122009559759792","7625597484987"],["64925062108545024","365203474360565760","924421294475182080","1386631941712773120","1364965817623511040","921351926895869952","431883715732439040","138819765771141120","29282294342350080","3660286792793760","205891132094649"],["3116402981210161152","19282743446237872128","54232715942544015360","91517708153043025920","102957421672173404160","81078969566836555776","45606920381345562624","18324209081790627840","5153683804253614080","966315713297552640","108710517745974672","5559060566555523"],["149587343098087735296","1009714565912092213248","3123804438290535284736","5857133321794753658880","7412934360396485099520","6671640924356836589568","4378264356609174011904","2110948886222280327168","742130467812520427520","185532616953130106880","31308629110840705536","3202018886335981248","150094635296999121"],["7180192468708211294208","52505157427428795088896","177204906317572183425024","365485119279992628314112","513963448987489633566720","520387992099833253986304","390290994074874940489728","219538684167117154025472","92617882383002549354496","28943088244688296673280","6512194855054866751488","999029892536826149376","93659052425327451504","4052555153018976267"],["344649238497994142121984","2714112753171703869210624","9923474753784042271801344","22327818196014095111553024","34538343771959303375683584","38855636743454216297644032","32784443502289495001137152","21075713680043246786445312","10373202826896285527703552","3889951060086107072888832","1094048735649217614249984","223782695928249057460224","31469441614910023705344","2723317062828752051424","109418989131512359209"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","75","5625","421875","31640625","2373046875","177978515625","13348388671875","1001129150390625","75084686279296875","5631351470947265625","422351360321044921875","31676352024078369140625","2375726401805877685546875","178179480135440826416015625"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["48","27"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193580,"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-d4d51d6562fb2476"]},{"id":193581,"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-3eb2f851fa4ef1f4","e-d4d51d6562fb2476"]},{"id":193582,"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-3eb2f851fa4ef1f4","e-d4d51d6562fb2476"]},{"id":193583,"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-d4d51d6562fb2476"]},{"id":193584,"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-3eb2f851fa4ef1f4"]}],"region_exact":{},"incoming_edges":[{"id":"e-3eb2f851fa4ef1f4","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-cdff05d2922e65c0","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":"48938e2efbfa18d9da4ce52636572e2b7f9adeeda68096bac5e2cf1cf59ca456","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-504bafdebe53d9d7","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^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-d4d51d6562fb2476","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-cdff05d2922e65c0","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":"56aaf0b100ca745b311ec9ed5ced96bb37812148e233f78038f1ac8f03352714","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-504bafdebe53d9d7","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^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-091e1093c43af59a","kind":"fingerprint_related","node_a_id":"n-cdff05d2922e65c0","node_b_id":"n-2d67e83eace19da6","fingerprint":"f6-f3c17ca3ebbc301e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c17ca3ebbc301e"},"other_id":"n-2d67e83eace19da6","other":{"id":"n-2d67e83eace19da6","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","first_index":0,"fingerprint":"f6-f3c17ca3ebbc301e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["324","1152","1024"],["5832","31104","55296","32768"],["104976","746496","1990656","2359296","1048576"]],"card_rows":[["324","1152","1024"],["5832","31104","55296","32768"],["104976","746496","1990656","2359296","1048576"],["1889568","16796160","59719680","106168320","94371840","33554432"]],"row_sums_preview":["2500","125000","6250000","312500000","15625000000","781250000000","39062500000000","1953125000000000"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7dcf11eddf2b9a8a","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-322a2e4c0bbd3231","e-1095a3d33bcc57bd"],"text":"A038321 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A038321 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-49a6561a67710e95","kind":"fingerprint_related","node_a_id":"n-cdff05d2922e65c0","node_b_id":"n-d5e838559a8c8a11","fingerprint":"f6-f3c17ca3ebbc301e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c17ca3ebbc301e"},"other_id":"n-d5e838559a8c8a11","other":{"id":"n-d5e838559a8c8a11","name":"shift_x_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-f3c17ca3ebbc301e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1024","1152","324"],["32768","55296","31104","5832"],["1048576","2359296","1990656","746496","104976"]],"card_rows":[["1024","1152","324"],["32768","55296","31104","5832"],["1048576","2359296","1990656","746496","104976"],["33554432","94371840","106168320","59719680","16796160","1889568"]],"row_sums_preview":["2500","125000","6250000","312500000","15625000000","781250000000","39062500000000","1953125000000000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A013610","basis A007318"],"edges":["e-89b2e37c644537dd","e-0f6f6478a7523569"],"text":"A038284 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A038284 \u2192 basis A013610 \u2192 basis A007318","chain":null}},{"id":"l-9e8f398f39b6a369","kind":"fingerprint_related","node_a_id":"n-cdff05d2922e65c0","node_b_id":"n-b2dfc4991f197cd3","fingerprint":"f6-f3c17ca3ebbc301e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c17ca3ebbc301e"},"other_id":"n-b2dfc4991f197cd3","other":{"id":"n-b2dfc4991f197cd3","name":"basis_map_A013610 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","first_index":0,"fingerprint":"f6-f3c17ca3ebbc301e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","9"],["256","288","81"]],"card_rows":[["1"],["16","9"],["256","288","81"],["4096","6912","3888","729"]],"row_sums_preview":["1","25","625","15625","390625","9765625","244140625","6103515625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-81e6fb4f9f226d77","steps":["basis A013610","basis A013610"],"edges":["e-00e3cfe53baaea9b","e-ce295775dae4b69c"],"text":"A038327 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038327 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-c2a8f31c58cc9aa4","kind":"fingerprint_related","node_a_id":"n-cdff05d2922e65c0","node_b_id":"n-824192ee04ae06b8","fingerprint":"f6-f3c17ca3ebbc301e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c17ca3ebbc301e"},"other_id":"n-824192ee04ae06b8","other":{"id":"n-824192ee04ae06b8","name":"basis_map_A013610 of basis_map_A013609 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","first_index":0,"fingerprint":"f6-f3c17ca3ebbc301e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["729","2592","2304"],["19683","104976","186624","110592"],["531441","3779136","10077696","11943936","5308416"]],"card_rows":[["729","2592","2304"],["19683","104976","186624","110592"],["531441","3779136","10077696","11943936","5308416"],["14348907","127545840","453496320","806215680","716636160","254803968"]],"row_sums_preview":["5625","421875","31640625","2373046875","177978515625","13348388671875","1001129150390625","75084686279296875"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-26633b980c3508e8","steps":["basis A013609","basis A013610"],"edges":["e-c8a2ee9cf24a53ed","e-487092da83f5a6cb"],"text":"A038226 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A038226 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-d057efe5aa70d352","kind":"fingerprint_related","node_a_id":"n-cdff05d2922e65c0","node_b_id":"n-6e6aac684bd07a75","fingerprint":"f6-f3c17ca3ebbc301e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c17ca3ebbc301e"},"other_id":"n-6e6aac684bd07a75","other":{"id":"n-6e6aac684bd07a75","name":"basis_map_A013611 of basis_map_A007318 of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-f3c17ca3ebbc301e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","16"],["81","288","256"]],"card_rows":[["1"],["9","16"],["81","288","256"],["729","3888","6912","4096"]],"row_sums_preview":["1","25","625","15625","390625","9765625","244140625","6103515625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2e0409c0014ec240","steps":["basis A007318","basis A013611"],"edges":["e-13c78408b476aecd","e-9b7b26e5d4ce6a74"],"text":"A013611 \u2192 basis A007318 \u2192 basis A013611"},"headline":"A013611 \u2192 basis A007318 \u2192 basis A013611","chain":null}},{"id":"l-e9785d6ae0569abb","kind":"fingerprint_related","node_a_id":"n-cdff05d2922e65c0","node_b_id":"n-44e3ef545532be70","fingerprint":"f6-f3c17ca3ebbc301e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3c17ca3ebbc301e"},"other_id":"n-44e3ef545532be70","other":{"id":"n-44e3ef545532be70","name":"basis_map_A013610 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","first_index":0,"fingerprint":"f6-f3c17ca3ebbc301e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","288","81"],["4096","6912","3888","729"],["65536","147456","124416","46656","6561"]],"card_rows":[["256","288","81"],["4096","6912","3888","729"],["65536","147456","124416","46656","6561"],["1048576","2949120","3317760","1866240","524880","59049"]],"row_sums_preview":["625","15625","390625","9765625","244140625","6103515625","152587890625","3814697265625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4397e742863fdbe2","steps":["x \u21a6 x+1","basis A013610"],"edges":["e-6056f3721ff524be","e-4676c68faaec715a"],"text":"A038305 \u2192 x \u21a6 x+1 \u2192 basis A013610"},"headline":"A038305 \u2192 x \u21a6 x+1 \u2192 basis A013610","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:               48,               27;\n  2:             2304,             2592,               729;\n  3:           110592,           186624,            104976,             19683;\n  4:          5308416,         11943936,          10077696,           3779136,            531441;\n  5:        254803968,        716636160,         806215680,         453496320,         127545840,         14348907;\n  6:      12230590464,      41278242816,       58047528960,       43535646720,       18366600960,       4132485216,        387420489;\n  7:     587068342272,    2311581597696,     3900793946112,     3656994324480,     2057059307520,     694257516288,     130173284304,     10460353203;\n  8:   28179280429056,  126806761930752,   249650812551168,   280857164120064,   197477693521920,   88864962084864,   24993270586368,   4016775629952,    282429536481;\n  9: 1352605460594688, 6847565144260608, 15407021574586368, 20221715816644608, 17062072720293888, 9597415905165312, 3599030964436992, 867623536069632, 122009559759792, 7625597484987;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (48+27*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[(48 + 27*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] = (48 + 27*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-f485cd65a8f8089f","steps":["basis A013610","basis A007318"],"edges":["e-b7b3e04e23eb02df","e-3eb2f851fa4ef1f4"],"text":"A038335 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A038335 \u2192 basis A013610 \u2192 basis A007318","chain":null}
