{"id":"n-4e07e76af0d9dd55","revision":0,"first_index":0,"fingerprint":"f6-1024c0779b1dc97d","name":"shift_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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":[["400","240","36"],["8000","7200","2160","216"],["160000","192000","86400","17280","1296"],["3200000","4800000","2880000","864000","129600","7776"],["64000000","115200000","86400000","34560000","7776000","933120","46656"],["1280000000","2688000000","2419200000","1209600000","362880000","65318400","6531840","279936"],["25600000000","61440000000","64512000000","38707200000","14515200000","3483648000","522547200","44789760","1679616"],["512000000000","1382400000000","1658880000000","1161216000000","522547200000","156764160000","31352832000","4031078400","302330880","10077696"],["10240000000000","30720000000000","41472000000000","33177600000000","17418240000000","6270566400000","1567641600000","268738560000","30233088000","2015539200","60466176"],["204800000000000","675840000000000","1013760000000000","912384000000000","547430400000000","229920768000000","68976230400000","14780620800000","2217093120000","221709312000","13302558720","362797056"],["4096000000000000","14745600000000000","24330240000000000","24330240000000000","16422912000000000","7882997760000000","2759049216000000","709469798400000","133025587200000","17736744960000","1596307046400","87071293440","2176782336"],["81920000000000000","319488000000000000","575078400000000000","632586240000000000","474439680000000000","256197427200000000","102478970880000000","30743691264000000","6917330534400000","1152888422400000","138346610688000","11319268147200","565963407360","13060694016"],["1638400000000000000","6881280000000000000","13418496000000000000","16102195200000000000","13284311040000000000","7970586624000000000","3586763980800000000","1229747650560000000","322808758272000000","64561751654400000","9684262748160000","1056465027072000","79234877030400","3656994324480","78364164096"],["32768000000000000000","147456000000000000000","309657600000000000000","402554880000000000000","362299392000000000000","239117598720000000000","119558799360000000000","46115536896000000000","13834661068800000000","3228087582720000000","581055764889600000","79234877030400000","7923487703040000","548549148672000","23509249228800","470184984576"],["655360000000000000000","3145728000000000000000","7077888000000000000000","9909043200000000000000","9661317120000000000000","6956148326400000000000","3825881579520000000000","1639663534080000000000","553386442752000000000","147569718067200000000","30989640794112000000","5071032129945600000","633879016243200000","58511909191680000","3761479876608000","150459195064320","2821109907456"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["676","17576","456976","11881376","308915776","8031810176","208827064576","5429503678976","141167095653376","3670344486987776","95428956661682176","2481152873203736576","64509974703297150976","1677259342285725925376","43608742899428874059776"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["400","240","36"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["20","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193300,"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-061f13a10a85161a"]},{"id":193301,"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-061f13a10a85161a","e-a2888f561fcb2920","e-c8e035344f807759"]},{"id":193302,"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-061f13a10a85161a","e-a2888f561fcb2920","e-c8e035344f807759"]},{"id":193303,"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-061f13a10a85161a"]},{"id":193304,"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-a2888f561fcb2920","e-c8e035344f807759"]}],"region_exact":{},"incoming_edges":[{"id":"e-a2888f561fcb2920","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-4e07e76af0d9dd55","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":"71f640e2d7cc9d0ef6c7d50fc15dfab58cca380a454b4de982eaf96a57d5a54e","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-7288f584ad081a07","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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-061f13a10a85161a","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-4e07e76af0d9dd55","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":"9d76fc95e366cb42993f39ea9117380009a1d09bd77562565d5959d1e82746aa","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-7288f584ad081a07","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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-c8e035344f807759","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-4e07e76af0d9dd55","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":"078d4065f4b003c55b12ddfcfd7ddb45f589adab53b21a07f98c082f83e8dd98","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-a2769223501d377d","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients in expansion of (1+5x)^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-118bedb7c0c0ff28","kind":"fingerprint_related","node_a_id":"n-a2769223501d377d","node_b_id":"n-4e07e76af0d9dd55","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-a2769223501d377d","other":{"id":"n-a2769223501d377d","name":"basis_map_A013611 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["36","240","400"],["216","2160","7200","8000"],["1296","17280","86400","192000","160000"]],"card_rows":[["36","240","400"],["216","2160","7200","8000"],["1296","17280","86400","192000","160000"],["7776","129600","864000","2880000","4800000","3200000"]],"row_sums_preview":["676","17576","456976","11881376","308915776","8031810176","208827064576","5429503678976"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["basis A013611"],"edges":["e-db3287eb465d1cf9"],"text":"A013612 \u2192 basis A013611"},"headline":"A013612 \u2192 basis A013611","chain":null}},{"id":"l-14b28aebc47ec631","kind":"fingerprint_related","node_a_id":"n-4e07e76af0d9dd55","node_b_id":"n-d1b3a560f252ca5d","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-d1b3a560f252ca5d","other":{"id":"n-d1b3a560f252ca5d","name":"derivative of basis_map_A013611 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["240","800"],["2160","14400","24000"],["17280","172800","576000","640000"]],"card_rows":[["240","800"],["2160","14400","24000"],["17280","172800","576000","640000"],["129600","1728000","8640000","19200000","16000000"]],"row_sums_preview":["1040","40560","1406080","45697600","1425765120","43248208640","1285089628160","37588871623680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["basis A013611","D"],"edges":["e-db3287eb465d1cf9","e-d22f3780bc70a92b"],"text":"A013612 \u2192 basis A013611 \u2192 D"},"headline":"A013612 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-212864f08aa0aae3","kind":"fingerprint_related","node_a_id":"n-fe2b68140d990ced","node_b_id":"n-4e07e76af0d9dd55","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-fe2b68140d990ced","other":{"id":"n-fe2b68140d990ced","name":"reversal of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","10"],["9","60","100"]],"card_rows":[["1"],["3","10"],["9","60","100"],["27","270","900","1000"]],"row_sums_preview":["1","13","169","2197","28561","371293","4826809","62748517"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013610","reverse"],"edges":["e-8d10edd099560dc3","e-654768579b54c833"],"text":"A038291 \u2192 basis A013610 \u2192 reverse"},"headline":"A038291 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-3a8db52d794bd99a","kind":"fingerprint_related","node_a_id":"n-6ca7f61b6ad83377","node_b_id":"n-4e07e76af0d9dd55","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-6ca7f61b6ad83377","other":{"id":"n-6ca7f61b6ad83377","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","3"],["100","60","9"]],"card_rows":[["1"],["10","3"],["100","60","9"],["1000","900","270","27"]],"row_sums_preview":["1","13","169","2197","28561","371293","4826809","62748517"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013610"],"edges":["e-8d10edd099560dc3"],"text":"A038291 \u2192 basis A013610"},"headline":"A038291 \u2192 basis A013610","chain":null}},{"id":"l-a7816bdadd385ad9","kind":"fingerprint_related","node_a_id":"n-6a5f903d0d9009ce","node_b_id":"n-4e07e76af0d9dd55","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-6a5f903d0d9009ce","other":{"id":"n-6a5f903d0d9009ce","name":"derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["60","18"]],"card_rows":[["3"],["60","18"],["900","540","81"],["12000","10800","3240","324"]],"row_sums_preview":["0","3","78","1521","26364","428415","6683274","101362989"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013610","D"],"edges":["e-8d10edd099560dc3","e-ec1a6e1208a933e0"],"text":"A038291 \u2192 basis A013610 \u2192 D"},"headline":"A038291 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-ab1751a8d10fe932","kind":"fingerprint_related","node_a_id":"n-4397e742863fdbe2","node_b_id":"n-4e07e76af0d9dd55","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-4397e742863fdbe2","other":{"id":"n-4397e742863fdbe2","name":"Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["100","60","9"],["1000","900","270","27"],["10000","12000","5400","1080","81"]],"card_rows":[["100","60","9"],["1000","900","270","27"],["10000","12000","5400","1080","81"],["100000","150000","90000","27000","4050","243"]],"row_sums_preview":["169","2197","28561","371293","4826809","62748517","815730721","10604499373"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4397e742863fdbe2","steps":[],"edges":[],"text":"A038305"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","chain":null}},{"id":"l-e05d0a58b858835f","kind":"fingerprint_related","node_a_id":"n-80a3aaf82f4475d0","node_b_id":"n-4e07e76af0d9dd55","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-80a3aaf82f4475d0","other":{"id":"n-80a3aaf82f4475d0","name":"reversal of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["9","60","100"],["27","270","900","1000"],["81","1080","5400","12000","10000"]],"card_rows":[["9","60","100"],["27","270","900","1000"],["81","1080","5400","12000","10000"],["243","4050","27000","90000","150000","100000"]],"row_sums_preview":["169","2197","28561","371293","4826809","62748517","815730721","10604499373"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4397e742863fdbe2","steps":["reverse"],"edges":["e-7b98e95757f40f31"],"text":"A038305 \u2192 reverse"},"headline":"A038305 \u2192 reverse","chain":null}},{"id":"l-f94075ad282563fe","kind":"fingerprint_related","node_a_id":"n-c3a62990f6820c27","node_b_id":"n-4e07e76af0d9dd55","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-c3a62990f6820c27","other":{"id":"n-c3a62990f6820c27","name":"shift_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["30","9"],["900","540","81"]],"card_rows":[["1"],["30","9"],["900","540","81"],["27000","24300","7290","729"]],"row_sums_preview":["1","39","1521","59319","2313441","90224199","3518743761","137231006679"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["x \u21a6 x+1","basis A007318"],"edges":["e-62ade03cd9f25f04","e-41449c7eb2a62420"],"text":"A038335 \u2192 x \u21a6 x+1 \u2192 basis A007318"},"headline":"A038335 \u2192 x \u21a6 x+1 \u2192 basis A007318","chain":null}},{"id":"l-fe1d8a1b0eb19831","kind":"fingerprint_related","node_a_id":"n-355f084f1a1b73e3","node_b_id":"n-4e07e76af0d9dd55","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-355f084f1a1b73e3","other":{"id":"n-355f084f1a1b73e3","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["60","18"],["900","540","81"],["12000","10800","3240","324"]],"card_rows":[["60","18"],["900","540","81"],["12000","10800","3240","324"],["150000","180000","81000","16200","1215"]],"row_sums_preview":["78","1521","26364","428415","6683274","101362989","1505964408","22024729467"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4397e742863fdbe2","steps":["D"],"edges":["e-04a8061d0e21fde2"],"text":"A038305 \u2192 D"},"headline":"A038305 \u2192 D","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:             400,             240,               36;\n  1:            8000,            7200,             2160,             216;\n  2:          160000,          192000,            86400,           17280,            1296;\n  3:         3200000,         4800000,          2880000,          864000,          129600,            7776;\n  4:        64000000,       115200000,         86400000,        34560000,         7776000,          933120,          46656;\n  5:      1280000000,      2688000000,       2419200000,      1209600000,       362880000,        65318400,        6531840,         279936;\n  6:     25600000000,     61440000000,      64512000000,     38707200000,     14515200000,      3483648000,      522547200,       44789760,       1679616;\n  7:    512000000000,   1382400000000,    1658880000000,   1161216000000,    522547200000,    156764160000,    31352832000,     4031078400,     302330880,     10077696;\n  8:  10240000000000,  30720000000000,   41472000000000,  33177600000000,  17418240000000,   6270566400000,  1567641600000,   268738560000,   30233088000,   2015539200,    60466176;\n  9: 204800000000000, 675840000000000, 1013760000000000, 912384000000000, 547430400000000, 229920768000000, 68976230400000, 14780620800000, 2217093120000, 221709312000, 13302558720, 362797056;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (20+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 240*x + 36*x^2.","mathematica":"P[0, x_] := 400 + 240*x + 36*x^2;\nP[n_, x_] := P[n, x] = Expand[(20 + 6*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] = 400 + 240*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 6*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-c11cb6bca0e3e7dc","steps":["basis A007318","x \u21a6 x+1"],"edges":["e-37fb8ca4839f71ce","e-061f13a10a85161a"],"text":"A038284 \u2192 basis A007318 \u2192 x \u21a6 x+1"},"headline":"A038284 \u2192 basis A007318 \u2192 x \u21a6 x+1","chain":null}
