{"id":"n-d19e6641dcda14df","revision":0,"first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","name":"shift_x_plus_one of basis_map_A013610 of Triangle of coefficients in expansion of (14 + x)^n.","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"],["18","3"],["324","108","9"],["5832","2916","486","27"],["104976","69984","17496","1944","81"],["1889568","1574640","524880","87480","7290","243"],["34012224","34012224","14171760","3149280","393660","26244","729"],["612220032","714256704","357128352","99202320","16533720","1653372","91854","2187"],["11019960576","14693280768","8571080448","2857026816","595213920","79361856","6613488","314928","6561"],["198359290368","297538935552","198359290368","77139724032","19284931008","3214155168","357128352","25509168","1062882","19683"],["3570467226624","5950778711040","4463084033280","1983592903680","578547930240","115709586048","16070775840","1530550080","95659380","3542940","59049"],["64268410079232","117825418478592","98187848732160","49093924366080","16364641455360","3818416339584","636402723264","75762228960","6313519080","350751060","11691702","177147"],["1156831381426176","2313662762852352","2120857532614656","1178254184785920","441845319294720","117825418478592","22910498037504","3272928291072","340930030320","25254076320","1262703816","38263752","531441"],["20822964865671168","45116423875620864","45116423875620864","27571147923990528","11487978301662720","3446393490498816","765865220110848","127644203351808","15955525418976","1477363464720","98490897648","4476858984","124357194","1594323"],["374813367582081024","874564524358189056","947444901388038144","631629934258692096","289497053201900544","96499017733966848","24124754433491712","4595191320665088","670132067596992","74459118621888","6204926551824","376056154656","15669006444","401769396","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","21","441","9261","194481","4084101","85766121","1801088541","37822859361","794280046581","16679880978201","350277500542221","7355827511386641","154472377739119461","3243919932521508681"],"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":[["18","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189740,"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-bcc80721f40fd9b6"]},{"id":189741,"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-bcc80721f40fd9b6","e-e65fa10f103aa9e3"]},{"id":189742,"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-bcc80721f40fd9b6","e-e65fa10f103aa9e3"]},{"id":189743,"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-bcc80721f40fd9b6"]},{"id":189744,"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-e65fa10f103aa9e3"]}],"region_exact":{},"incoming_edges":[{"id":"e-e65fa10f103aa9e3","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-d19e6641dcda14df","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":"4821db389afe204f7ffdce281378da4c2640ed87b17229cca2b60af4dd220109","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-804e872f93921cf0","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (14 + x)^n.","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-bcc80721f40fd9b6","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-d19e6641dcda14df","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":"8e5b627eebb34eba70f264351daf454c8e16bd989c713ff2f29f7973b747eb8e","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-804e872f93921cf0","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (14 + x)^n.","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-20079656da066976","kind":"fingerprint_related","node_a_id":"n-d19e6641dcda14df","node_b_id":"n-a927649f3968c00b","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-a927649f3968c00b","other":{"id":"n-a927649f3968c00b","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["48","8"],["864","288","24"],["13824","6912","1152","64"]],"card_rows":[["48","8"],["864","288","24"],["13824","6912","1152","64"],["207360","138240","34560","3840","160"]],"row_sums_preview":["56","1176","21952","384160","6453888","105413504","1686616064","26564203008"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["D"],"edges":["e-65f122626ee17875"],"text":"A038328 \u2192 D"},"headline":"A038328 \u2192 D","chain":null}},{"id":"l-22e5380968604b19","kind":"fingerprint_related","node_a_id":"n-efce1b6c07b70e1a","node_b_id":"n-d19e6641dcda14df","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-efce1b6c07b70e1a","other":{"id":"n-efce1b6c07b70e1a","name":"Triangle of coefficients in expansion of (1+6x)^n.","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","6"],["1","12","36"]],"card_rows":[["1"],["1","6"],["1","12","36"],["1","18","108","216"]],"row_sums_preview":["1","7","49","343","2401","16807","117649","823543"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A013613"],"derivation":{"root":"n-efce1b6c07b70e1a","steps":[],"edges":[],"text":"A013613"},"headline":"A013613: Triangle of coefficients in expansion of (1+6x)^n.","chain":null}},{"id":"l-51c4b896800901d5","kind":"fingerprint_related","node_a_id":"n-64bc2cdf1204163e","node_b_id":"n-d19e6641dcda14df","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-64bc2cdf1204163e","other":{"id":"n-64bc2cdf1204163e","name":"shift_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","2"],["144","48","4"]],"card_rows":[["1"],["12","2"],["144","48","4"],["1728","864","144","8"]],"row_sums_preview":["1","14","196","2744","38416","537824","7529536","105413504"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013609","basis A007318"],"edges":["e-88ca34e9877f4f47","e-12984afc1d27ca59"],"text":"A038291 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A038291 \u2192 basis A013609 \u2192 basis A007318","chain":null}},{"id":"l-521b2096dfe7683d","kind":"fingerprint_related","node_a_id":"n-22693abd431a71f2","node_b_id":"n-d19e6641dcda14df","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-22693abd431a71f2","other":{"id":"n-22693abd431a71f2","name":"Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","1"],["36","12","1"]],"card_rows":[["1"],["6","1"],["36","12","1"],["216","108","18","1"]],"row_sums_preview":["1","7","49","343","2401","16807","117649","823543"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A038255"],"derivation":{"root":"n-22693abd431a71f2","steps":[],"edges":[],"text":"A038255"},"headline":"A038255: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","chain":null}},{"id":"l-5b9962cfa9110c9f","kind":"fingerprint_related","node_a_id":"n-d19e6641dcda14df","node_b_id":"n-72192ea1b98c5050","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-72192ea1b98c5050","other":{"id":"n-72192ea1b98c5050","name":"reversal of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["4","48","144"],["8","144","864","1728"],["16","384","3456","13824","20736"]],"card_rows":[["4","48","144"],["8","144","864","1728"],["16","384","3456","13824","20736"],["32","960","11520","69120","207360","248832"]],"row_sums_preview":["196","2744","38416","537824","7529536","105413504","1475789056","20661046784"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["reverse"],"edges":["e-896f56ac2a40747d"],"text":"A038328 \u2192 reverse"},"headline":"A038328 \u2192 reverse","chain":null}},{"id":"l-6da7ddac57d021b9","kind":"fingerprint_related","node_a_id":"n-0817f7209fcd849d","node_b_id":"n-d19e6641dcda14df","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-0817f7209fcd849d","other":{"id":"n-0817f7209fcd849d","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["12","2"]],"card_rows":[["1"],["12","2"],["108","36","3"],["864","432","72","4"]],"row_sums_preview":["0","1","14","147","1372","12005","100842","823543"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["D"],"edges":["e-e34388594b9483c9"],"text":"A038255 \u2192 D"},"headline":"A038255 \u2192 D","chain":null}},{"id":"l-cf298d5d5e21703a","kind":"fingerprint_related","node_a_id":"n-8ff9f44decd2285a","node_b_id":"n-d19e6641dcda14df","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-8ff9f44decd2285a","other":{"id":"n-8ff9f44decd2285a","name":"derivative of Triangle of coefficients in expansion of (1+6x)^n.","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["6"],["12","72"]],"card_rows":[["6"],["12","72"],["18","216","648"],["24","432","2592","5184"]],"row_sums_preview":["0","6","84","882","8232","72030","605052","4941258"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efce1b6c07b70e1a","steps":["D"],"edges":["e-b1a99f7ec702dba2"],"text":"A013613 \u2192 D"},"headline":"A013613 \u2192 D","chain":null}},{"id":"l-ddfb574d92dbfe1e","kind":"fingerprint_related","node_a_id":"n-d19e6641dcda14df","node_b_id":"n-95ff44822e25b07f","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-95ff44822e25b07f","other":{"id":"n-95ff44822e25b07f","name":"shift_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["324","108","9"],["5832","2916","486","27"],["104976","69984","17496","1944","81"]],"card_rows":[["324","108","9"],["5832","2916","486","27"],["104976","69984","17496","1944","81"],["1889568","1574640","524880","87480","7290","243"]],"row_sums_preview":["441","9261","194481","4084101","85766121","1801088541","37822859361","794280046581"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-5083e5230277d126","e-192fdbcaf1e01247"],"text":"A038329 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"A038329 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-e76a31eb9bd9f9ee","kind":"fingerprint_related","node_a_id":"n-155eef1a24bcc703","node_b_id":"n-d19e6641dcda14df","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-155eef1a24bcc703","other":{"id":"n-155eef1a24bcc703","name":"mobius_x_over_x_plus_one of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","12","36"],["1","18","108","216"],["1","24","216","864","1296"]],"card_rows":[["1","12","36"],["1","18","108","216"],["1","24","216","864","1296"],["1","30","360","2160","6480","7776"]],"row_sums_preview":["49","343","2401","16807","117649","823543","5764801","40353607"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["x \u21a6 x/(x+1)"],"edges":["e-43aab10e23af0f21"],"text":"A013612 \u2192 x \u21a6 x/(x+1)"},"headline":"A013612 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-f000fac3c99f291c","kind":"fingerprint_related","node_a_id":"n-ea9b82d727693fb6","node_b_id":"n-d19e6641dcda14df","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-ea9b82d727693fb6","other":{"id":"n-ea9b82d727693fb6","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","48","4"],["1728","864","144","8"],["20736","13824","3456","384","16"]],"card_rows":[["144","48","4"],["1728","864","144","8"],["20736","13824","3456","384","16"],["248832","207360","69120","11520","960","32"]],"row_sums_preview":["196","2744","38416","537824","7529536","105413504","1475789056","20661046784"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":[],"edges":[],"text":"A038328"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","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:           18,            3;\n  2:          324,          108,            9;\n  3:         5832,         2916,          486,          27;\n  4:       104976,        69984,        17496,        1944,          81;\n  5:      1889568,      1574640,       524880,       87480,        7290,        243;\n  6:     34012224,     34012224,     14171760,     3149280,      393660,      26244,       729;\n  7:    612220032,    714256704,    357128352,    99202320,    16533720,    1653372,     91854,     2187;\n  8:  11019960576,  14693280768,   8571080448,  2857026816,   595213920,   79361856,   6613488,   314928,    6561;\n  9: 198359290368, 297538935552, 198359290368, 77139724032, 19284931008, 3214155168, 357128352, 25509168, 1062882, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (18+3*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[(18 + 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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (18 + 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-a679863398df5985","steps":["basis A013610","x \u21a6 x+1"],"edges":["e-011a72a323f5e83e","e-bcc80721f40fd9b6"],"text":"A147716 \u2192 basis A013610 \u2192 x \u21a6 x+1"},"headline":"A147716 \u2192 basis A013610 \u2192 x \u21a6 x+1","chain":null}
