{"id":"n-855d7b8331137387","revision":0,"first_index":0,"fingerprint":"f6-48a0cbb267bcb941","name":"reversal of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^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":[["49","168","144"],["343","1764","3024","1728"],["2401","16464","42336","48384","20736"],["16807","144060","493920","846720","725760","248832"],["117649","1210104","5186160","11854080","15240960","10450944","2985984"],["823543","9882516","50824368","145212480","248935680","256048128","146313216","35831808"],["5764801","79060128","474360768","1626379776","3485099520","4779565056","4096770048","2006581248","429981696"],["40353607","622598508","4269246912","17076987648","43912253952","75278149632","86032171008","63207309312","27088846848","5159780352"],["282475249","4842432840","37355910480","170769876480","512309629440","1053894094848","1505562992640","1474837217280","948109639680","361184624640","61917364224"],["1977326743","37286732868","319600567440","1643660061120","5635405923840","13524974217216","23185670086656","28390616432640","24334814085120","13905608048640","4767637045248","743008370688"],["13841287201","284735050992","2684644766496","15340827237120","59171762200320","162299690606592","324599381213184","476962356068352","511031095787520","389357025361920","200240755900416","62412703137792","8916100448256"],["96889010407","2159240803356","22209333977376","139601527857792","598292262247680","1846158980649984","4219791955771392","7233929067036672","9300765943332864","8857872326983680","6073969595645952","2839777992769536","811365140791296","106993205379072"],["678223072849","16277353748376","181376227481904","1243722702733056","5863264170027264","20102620011522048","51692451458199552","101275006938513408","151912510407770112","173614297608880128","148812255093325824","92766081097138176","39756891898773504","10485334127149056","1283918464548864"],["4747561509943","122080153112820","1464961837353840","10882573648914240","55967521622987520","211077510120981504","603078600345661440","1329234466067988480","2278687656116551680","3038250208155402240","3125057356959842304","2435109628799877120","1391491216457072640","550480041675325440","134811438777630720","15407021574586368"],["33232930569601","911531809909056","11719694698830720","93757557590645760","522363535147883520","2149152830322720768","6754480323871408128","16541584466623856640","31901627185631723520","48612003330486435840","58334403996583723008","54546455685117247488","38961754060798033920","20551254889212149760","7549440571547320320","1725586416353673216","184884258895036416"]],"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:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["49","168","144"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["7","12"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193030,"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-ba38a88165297efb"]},{"id":193031,"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-ba38a88165297efb"]},{"id":193032,"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-ba38a88165297efb"]}],"region_exact":{},"incoming_edges":[{"id":"e-ba38a88165297efb","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":2,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-855d7b8331137387","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"fc30f5e96d21870e9f255f7c31d650c8ec7375e3679878c9d10460dd9e259fcf","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-5cf6acfdac607d6d","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^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-3db54f0362a40874","kind":"fingerprint_related","node_a_id":"n-5cf6acfdac607d6d","node_b_id":"n-855d7b8331137387","fingerprint":"f6-48a0cbb267bcb941","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-48a0cbb267bcb941"},"other_id":"n-5cf6acfdac607d6d","other":{"id":"n-5cf6acfdac607d6d","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.","first_index":0,"fingerprint":"f6-48a0cbb267bcb941","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","168","49"],["1728","3024","1764","343"],["20736","48384","42336","16464","2401"]],"card_rows":[["144","168","49"],["1728","3024","1764","343"],["20736","48384","42336","16464","2401"],["248832","725760","846720","493920","144060","16807"]],"row_sums_preview":["361","6859","130321","2476099","47045881","893871739","16983563041","322687697779"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5cf6acfdac607d6d","steps":[],"edges":[],"text":"A038333"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.","chain":null}},{"id":"l-3f96ff4dd6015690","kind":"fingerprint_related","node_a_id":"n-0b6560aa9867a4e4","node_b_id":"n-855d7b8331137387","fingerprint":"f6-48a0cbb267bcb941","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-48a0cbb267bcb941"},"other_id":"n-0b6560aa9867a4e4","other":{"id":"n-0b6560aa9867a4e4","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-48a0cbb267bcb941","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["196","672","576"],["2744","14112","24192","13824"],["38416","263424","677376","774144","331776"]],"card_rows":[["196","672","576"],["2744","14112","24192","13824"],["38416","263424","677376","774144","331776"],["537824","4609920","15805440","27095040","23224320","7962624"]],"row_sums_preview":["1444","54872","2085136","79235168","3010936384","114415582592","4347792138496","165216101262848"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A013611"],"edges":["e-06104b6211fa5aaa"],"text":"A038284 \u2192 basis A013611"},"headline":"A038284 \u2192 basis A013611","chain":null}},{"id":"l-44a8b301a873322d","kind":"fingerprint_related","node_a_id":"n-855d7b8331137387","node_b_id":"n-9f4aefa5f442e451","fingerprint":"f6-48a0cbb267bcb941","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-48a0cbb267bcb941"},"other_id":"n-9f4aefa5f442e451","other":{"id":"n-9f4aefa5f442e451","name":"basis_map_A013610 of basis_map_A013609 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-48a0cbb267bcb941","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["21","36"],["147","504","432"]],"card_rows":[["3"],["21","36"],["147","504","432"],["1029","5292","9072","5184"]],"row_sums_preview":["3","57","1083","20577","390963","7428297","141137643","2681615217"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A013611","basis A013610"],"edges":["e-1a05d30902084644","e-44f7d78af53a1cc4"],"text":"A120910 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A120910 \u2192 basis A013611 \u2192 basis A013610","chain":null}},{"id":"l-455ec488683b91e9","kind":"fingerprint_related","node_a_id":"n-96d5ef3a0d05e14f","node_b_id":"n-855d7b8331137387","fingerprint":"f6-48a0cbb267bcb941","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-48a0cbb267bcb941"},"other_id":"n-96d5ef3a0d05e14f","other":{"id":"n-96d5ef3a0d05e14f","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+6x)^n.","first_index":0,"fingerprint":"f6-48a0cbb267bcb941","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["7","12"],["49","168","144"]],"card_rows":[["1"],["7","12"],["49","168","144"],["343","1764","3024","1728"]],"row_sums_preview":["1","19","361","6859","130321","2476099","47045881","893871739"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["x \u21a6 x/(x+1)"],"edges":["e-05b85140032a040d"],"text":"A038271 \u2192 x \u21a6 x/(x+1)"},"headline":"A038271 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-6549abb2fa5f57b7","kind":"fingerprint_related","node_a_id":"n-0ea2fa864539dca6","node_b_id":"n-855d7b8331137387","fingerprint":"f6-48a0cbb267bcb941","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-48a0cbb267bcb941"},"other_id":"n-0ea2fa864539dca6","other":{"id":"n-0ea2fa864539dca6","name":"derivative of basis_map_A013609 of Triangle of coefficients in expansion of (1+6x)^n.","first_index":0,"fingerprint":"f6-48a0cbb267bcb941","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["12"],["168","288"]],"card_rows":[["12"],["168","288"],["1764","6048","5184"],["16464","84672","145152","82944"]],"row_sums_preview":["0","12","456","12996","329232","7819260","178279128","3951854004"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["x \u21a6 x/(x+1)","D"],"edges":["e-05b85140032a040d","e-b662ccb71cbbc6ed"],"text":"A038271 \u2192 x \u21a6 x/(x+1) \u2192 D"},"headline":"A038271 \u2192 x \u21a6 x/(x+1) \u2192 D","chain":null}},{"id":"l-7121d1d9bb8bd1e8","kind":"fingerprint_related","node_a_id":"n-1afcf9ef6685c385","node_b_id":"n-855d7b8331137387","fingerprint":"f6-48a0cbb267bcb941","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-48a0cbb267bcb941"},"other_id":"n-1afcf9ef6685c385","other":{"id":"n-1afcf9ef6685c385","name":"reversal of basis_map_A013609 of Triangle of coefficients in expansion of (1+6x)^n.","first_index":0,"fingerprint":"f6-48a0cbb267bcb941","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","7"],["144","168","49"]],"card_rows":[["1"],["12","7"],["144","168","49"],["1728","3024","1764","343"]],"row_sums_preview":["1","19","361","6859","130321","2476099","47045881","893871739"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["x \u21a6 x/(x+1)","reverse"],"edges":["e-05b85140032a040d","e-4f946466255d4ca9"],"text":"A038271 \u2192 x \u21a6 x/(x+1) \u2192 reverse"},"headline":"A038271 \u2192 x \u21a6 x/(x+1) \u2192 reverse","chain":null}},{"id":"l-9ba8a0b5ecad9fcb","kind":"fingerprint_related","node_a_id":"n-ec06bf4fbcbe06db","node_b_id":"n-855d7b8331137387","fingerprint":"f6-48a0cbb267bcb941","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-48a0cbb267bcb941"},"other_id":"n-ec06bf4fbcbe06db","other":{"id":"n-ec06bf4fbcbe06db","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.","first_index":0,"fingerprint":"f6-48a0cbb267bcb941","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["168","98"],["3024","3528","1029"],["48384","84672","49392","9604"]],"card_rows":[["168","98"],["3024","3528","1029"],["48384","84672","49392","9604"],["725760","1693440","1481760","576240","84035"]],"row_sums_preview":["266","7581","192052","4561235","103996158","2305248169","50056817384","1069964471583"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5cf6acfdac607d6d","steps":["D"],"edges":["e-4b5315c96a17476a"],"text":"A038333 \u2192 D"},"headline":"A038333 \u2192 D","chain":null}},{"id":"l-a3d39420f25d4cca","kind":"fingerprint_related","node_a_id":"n-537d24a405b4f361","node_b_id":"n-855d7b8331137387","fingerprint":"f6-48a0cbb267bcb941","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-48a0cbb267bcb941"},"other_id":"n-537d24a405b4f361","other":{"id":"n-537d24a405b4f361","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-48a0cbb267bcb941","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["21","36"],["441","1512","1296"]],"card_rows":[["1"],["21","36"],["441","1512","1296"],["9261","47628","81648","46656"]],"row_sums_preview":["1","57","3249","185193","10556001","601692057","34296447249","1954897493193"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["basis A013611"],"edges":["e-3428d0ea6005b0ef"],"text":"A038335 \u2192 basis A013611"},"headline":"A038335 \u2192 basis A013611","chain":null}},{"id":"l-d746bb999e3e1e08","kind":"fingerprint_related","node_a_id":"n-855d7b8331137387","node_b_id":"n-23a8ee9d20fa8cdf","fingerprint":"f6-48a0cbb267bcb941","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-48a0cbb267bcb941"},"other_id":"n-23a8ee9d20fa8cdf","other":{"id":"n-23a8ee9d20fa8cdf","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-48a0cbb267bcb941","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["672","1152"],["14112","48384","41472"],["263424","1354752","2322432","1327104"]],"card_rows":[["672","1152"],["14112","48384","41472"],["263424","1354752","2322432","1327104"],["4609920","31610880","81285120","92897280","39813120"]],"row_sums_preview":["1824","103968","5267712","250216320","11409864192","505837312512","21967791857664","939123101915136"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A013611","D"],"edges":["e-06104b6211fa5aaa","e-ef8d9c0a8297931c"],"text":"A038284 \u2192 basis A013611 \u2192 D"},"headline":"A038284 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-ec0fb50b1e45959d","kind":"fingerprint_related","node_a_id":"n-855d7b8331137387","node_b_id":"n-76d494d4a790974a","fingerprint":"f6-48a0cbb267bcb941","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-48a0cbb267bcb941"},"other_id":"n-76d494d4a790974a","other":{"id":"n-76d494d4a790974a","name":"reversal of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-48a0cbb267bcb941","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["576","672","196"],["13824","24192","14112","2744"],["331776","774144","677376","263424","38416"]],"card_rows":[["576","672","196"],["13824","24192","14112","2744"],["331776","774144","677376","263424","38416"],["7962624","23224320","27095040","15805440","4609920","537824"]],"row_sums_preview":["1444","54872","2085136","79235168","3010936384","114415582592","4347792138496","165216101262848"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A013611","reverse"],"edges":["e-06104b6211fa5aaa","e-5a5273c72b13fe2e"],"text":"A038284 \u2192 basis A013611 \u2192 reverse"},"headline":"A038284 \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:         49,         168,          144;\n  1:        343,        1764,         3024,          1728;\n  2:       2401,       16464,        42336,         48384,         20736;\n  3:      16807,      144060,       493920,        846720,        725760,         248832;\n  4:     117649,     1210104,      5186160,      11854080,      15240960,       10450944,        2985984;\n  5:     823543,     9882516,     50824368,     145212480,     248935680,      256048128,      146313216,       35831808;\n  6:    5764801,    79060128,    474360768,    1626379776,    3485099520,     4779565056,     4096770048,     2006581248,      429981696;\n  7:   40353607,   622598508,   4269246912,   17076987648,   43912253952,    75278149632,    86032171008,    63207309312,    27088846848,     5159780352;\n  8:  282475249,  4842432840,  37355910480,  170769876480,  512309629440,  1053894094848,  1505562992640,  1474837217280,   948109639680,   361184624640,   61917364224;\n  9: 1977326743, 37286732868, 319600567440, 1643660061120, 5635405923840, 13524974217216, 23185670086656, 28390616432640, 24334814085120, 13905608048640, 4767637045248, 743008370688;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (7+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 49 + 168*x + 144*x^2.","mathematica":"P[0, x_] := 49 + 168*x + 144*x^2;\nP[n_, x_] := P[n, x] = Expand[(7 + 12*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] = 49 + 168*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (7 + 12*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-5cf6acfdac607d6d","steps":["reverse"],"edges":["e-ba38a88165297efb"],"text":"A038333 \u2192 reverse"},"headline":"A038333 \u2192 reverse","chain":null}