{"id":"n-ce03950bfb372f88","revision":0,"first_index":0,"fingerprint":"f6-9111d60e4a92b32c","name":"basis_map_A013610 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(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":[["196","252","81"],["2744","5292","3402","729"],["38416","98784","95256","40824","6561"],["537824","1728720","2222640","1428840","459270","59049"],["7529536","29042496","46675440","40007520","19289340","4960116","531441"],["105413504","474360768","914838624","980184240","630118440","243045684","52081218","4782969"],["1475789056","7589772288","17076987648","21956126976","17643316320","9073705536","2916548208","535692528","43046721"],["20661046784","119538913536","307385777664","461078666496","444611571264","285821724384","122495024736","33748629264","5423886846","387420489"],["289254654976","1859494210560","5379251109120","9221573329920","10374269996160","8003008282752","4287325865760","1574936032320","379672079220","54238868460","3486784401"],["4049565169664","28636210842624","92044963422720","177515286600960","228233939915520","205410545923968","132049636665408","60635037244320","19489833399960","4176392871420","536964797754","31381059609"],["56693912375296","437353038323712","1546355385501696","3313618683217920","4792912738225920","4929853102175232","3697389826631424","2037337251409152","818573002798320","233878000799520","45105043011336","5272018014312","282429536481"],["793714773254144","6633187747909632","25585152741937152","60307860034566144","96923346484124160","112154158074486528","96132135492417024","61799229959410944","29796057301858848","10641449036378160","2736372609354384","479753639302392","51402175639542","2541865828329"],["11112006825558016","100008061430022144","417890828118306816","1074576415161360384","1899697591088833536","2442468331399928832","2355237319564217088","1730378438863506432","973337871860722368","417144802226023872","134082257858364816","31343904434422944","5037413212675116","498205702352484","22876792454961"],["155568095557812224","1500120921450332160","6750544146526494720","18805087265323806720","36266954011695912960","51291834959398505472","54955537456498398720","45422434020167043840","29200136155821671040","14600068077910835520","5631454830051322272","1645554982807204560","352618924887258120","52311598747010820","4804126415541810","205891132094649"],["2177953337809371136","22401805760324960256","108008706344423915520","324026119033271746560","676983141551657041920","1044488275536842293248","1231004039025564131328","1130513913390824202240","817603812363006789120","467202178493146736640","210240980321916031488","73720863229762764288","19746659793686454720","3905932706443474560","538062158540682720","46119613589201376","1853020188851841"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["529","12167","279841","6436343","148035889","3404825447","78310985281","1801152661463","41426511213649","952809757913927","21914624432020321","504036361936467383","11592836324538749809","266635235464391245607","6132610415680998648961"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["196","252","81"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["14","9"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203765,"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-066880bf2751c810"]},{"id":203766,"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-066880bf2751c810"]},{"id":203767,"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-066880bf2751c810"]}],"region_exact":{},"incoming_edges":[{"id":"e-066880bf2751c810","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-ce03950bfb372f88","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":"030b10bfb5cb943bbaacbd1d8be7f17a8cb8818c244654dcd9ea205bdbe164c7","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-809ca598570cd420","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(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-64fa174d6fa2aed6","kind":"fingerprint_related","node_a_id":"n-6d0748781e5e30ed","node_b_id":"n-ce03950bfb372f88","fingerprint":"f6-9111d60e4a92b32c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9111d60e4a92b32c"},"other_id":"n-6d0748781e5e30ed","other":{"id":"n-6d0748781e5e30ed","name":"shift_x_plus_one of basis_map_A013610 of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-9111d60e4a92b32c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["14","9"],["196","252","81"]],"card_rows":[["1"],["14","9"],["196","252","81"],["2744","5292","3402","729"]],"row_sums_preview":["1","23","529","12167","279841","6436343","148035889","3404825447"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["basis A013610","basis A013610"],"edges":["e-254f1b828fe6620e","e-08a03f724769d7c5"],"text":"A038303 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038303 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-8f86aec23b66c47f","kind":"fingerprint_related","node_a_id":"n-4b2df01c2184142e","node_b_id":"n-ce03950bfb372f88","fingerprint":"f6-9111d60e4a92b32c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9111d60e4a92b32c"},"other_id":"n-4b2df01c2184142e","other":{"id":"n-4b2df01c2184142e","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","first_index":0,"fingerprint":"f6-9111d60e4a92b32c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["324","1008","784"],["5832","27216","42336","21952"],["104976","653184","1524096","1580544","614656"]],"card_rows":[["324","1008","784"],["5832","27216","42336","21952"],["104976","653184","1524096","1580544","614656"],["1889568","14696640","45722880","71124480","55319040","17210368"]],"row_sums_preview":["2116","97336","4477456","205962976","9474296896","435817657216","20047612231936","922190162669056"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7dcf11eddf2b9a8a","steps":["basis A013611"],"edges":["e-e460f89b751823ed"],"text":"A038321 \u2192 basis A013611"},"headline":"A038321 \u2192 basis A013611","chain":null}},{"id":"l-97485eb36d334c9c","kind":"fingerprint_related","node_a_id":"n-1bb976829c17ec98","node_b_id":"n-ce03950bfb372f88","fingerprint":"f6-9111d60e4a92b32c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9111d60e4a92b32c"},"other_id":"n-1bb976829c17ec98","other":{"id":"n-1bb976829c17ec98","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","first_index":0,"fingerprint":"f6-9111d60e4a92b32c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1008","1568"],["27216","84672","65856"],["653184","3048192","4741632","2458624"]],"card_rows":[["1008","1568"],["27216","84672","65856"],["653184","3048192","4741632","2458624"],["14696640","91445760","213373440","221276160","86051840"]],"row_sums_preview":["2576","177744","10901632","626843840","34601779968","1856962191616","97623155216384","5051998282447872"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7dcf11eddf2b9a8a","steps":["basis A013611","D"],"edges":["e-e460f89b751823ed","e-27550b5f4421b741"],"text":"A038321 \u2192 basis A013611 \u2192 D"},"headline":"A038321 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-d0c764bebb7469e7","kind":"fingerprint_related","node_a_id":"n-8c08995d48487910","node_b_id":"n-ce03950bfb372f88","fingerprint":"f6-9111d60e4a92b32c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9111d60e4a92b32c"},"other_id":"n-8c08995d48487910","other":{"id":"n-8c08995d48487910","name":"reversal of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","first_index":0,"fingerprint":"f6-9111d60e4a92b32c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["784","1008","324"],["21952","42336","27216","5832"],["614656","1580544","1524096","653184","104976"]],"card_rows":[["784","1008","324"],["21952","42336","27216","5832"],["614656","1580544","1524096","653184","104976"],["17210368","55319040","71124480","45722880","14696640","1889568"]],"row_sums_preview":["2116","97336","4477456","205962976","9474296896","435817657216","20047612231936","922190162669056"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7dcf11eddf2b9a8a","steps":["basis A013611","reverse"],"edges":["e-e460f89b751823ed","e-541bf93e05058bc8"],"text":"A038321 \u2192 basis A013611 \u2192 reverse"},"headline":"A038321 \u2192 basis A013611 \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:           196,            252,             81;\n  1:          2744,           5292,           3402,             729;\n  2:         38416,          98784,          95256,           40824,            6561;\n  3:        537824,        1728720,        2222640,         1428840,          459270,           59049;\n  4:       7529536,       29042496,       46675440,        40007520,        19289340,         4960116,          531441;\n  5:     105413504,      474360768,      914838624,       980184240,       630118440,       243045684,        52081218,        4782969;\n  6:    1475789056,     7589772288,    17076987648,     21956126976,     17643316320,      9073705536,      2916548208,      535692528,       43046721;\n  7:   20661046784,   119538913536,   307385777664,    461078666496,    444611571264,    285821724384,    122495024736,    33748629264,     5423886846,     387420489;\n  8:  289254654976,  1859494210560,  5379251109120,   9221573329920,  10374269996160,   8003008282752,   4287325865760,  1574936032320,   379672079220,   54238868460,   3486784401;\n  9: 4049565169664, 28636210842624, 92044963422720, 177515286600960, 228233939915520, 205410545923968, 132049636665408, 60635037244320, 19489833399960, 4176392871420, 536964797754, 31381059609;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (14+9*x)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 252*x + 81*x^2.","mathematica":"P[0, x_] := 196 + 252*x + 81*x^2;\nP[n_, x_] := P[n, x] = Expand[(14 + 9*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] = 196 + 252*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = (14 + 9*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-108b34f10b7468f4","steps":["x \u21a6 x+1","basis A013610"],"edges":["e-3a9169eda66e0655","e-066880bf2751c810"],"text":"A038281 \u2192 x \u21a6 x+1 \u2192 basis A013610"},"headline":"A038281 \u2192 x \u21a6 x+1 \u2192 basis A013610","chain":null}