{"id":"n-6a03653d7a50bc7a","revision":0,"first_index":0,"fingerprint":"f6-6bae7ffb6b8d3725","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["198","242"],["2673","6534","3993"],["32076","117612","143748","58564"],["360855","1764180","3234330","2635380","805255"],["3897234","23816430","58217940","71155260","43483770","10629366"],["40920957","300087018","916932555","1494260460","1369738755","669650058","136410197"],["420901272","3601044216","13203828792","26896688280","32873730120","24107402088","9821534184","1714871048"],["4261625379","41669225928","178251688692","435726350136","665693034930","650899856376","397772134452","138904554888","21221529219"],["42616253790","468778791690","2291807426040","6535895252040","11982474628740","14645246768460","11933164033560","6250704969960","1909937629710","259374246010"],["421900912521","5156566708590","28361116897245","92436232850280","197710831374210","289975886015508","295345809830610","206273264008680","94541912670645","25678050354990","3138428376721"],["4142299868388","55690920452772","340333402766940","1247889143478780","3050395684059240","5219565948279144","6379469492341176","5569378128234360","3403508856143220","1386614719169460","338950264685868","37661140520652"],["40387423716783","592348881179484","3981900812373198","16222558865224140","44612036879366385","87241316564094264","124399655100652932","130323448200684024","99552634042189185","54077974047608940","19828590484123278","4406353440916284","448795257871103"],["391447337562666","6219663252384582","45610863850820268","204404241701824164","624568516311129390","1374050735884484658","2239193791811752776","2736792412214364504","2508726377863167462","1703456182499681610","832800800333177676","277600266777725892","56548202491758978","5316497670165374"],["3774670755068565","64588810697839890","513122218321728015","2508597511795114740","8431674970200246765","20610761038267269870","37786395236823328095","52780996521277029720","56446343501921267895","45993316927491403470","28107027011244746565","12492012004997665140","3817003668193731015","717727185472325490","62658722541234765"],["36236839248658224","664342052892067440","5683815341409910320","30103170141541376880","110378290518985048560","296794958951048686128","604582323789173249520","950057937382986534960","1161181923468094653840","1103839606259793683280","809482377923848701072","449712432179915945040","183216176073299088720","51676357354007435280","9022856045937806160","735195677817154576"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["440","13200","352000","8800000","211200000","4928000000","112640000000","2534400000000","56320000000000","1239040000000000","27033600000000000","585728000000000000","12615680000000000000","270336000000000000000","5767168000000000000000"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["198","242"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["18","22"],["9","11"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193377,"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-4169abb442691e46"]},{"id":193378,"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-4169abb442691e46"]},{"id":193379,"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-4169abb442691e46"]},{"id":193380,"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-4169abb442691e46"]}],"region_exact":{},"incoming_edges":[{"id":"e-4169abb442691e46","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-6a03653d7a50bc7a","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"a58a1c50c478feaeb167601303dafddc5d8c89f1194cc2648eafc64f4556e9cc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-cfc5b48d32898a3e","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^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-267958b3cb19c12b","kind":"fingerprint_related","node_a_id":"n-4113d906b9d95b00","node_b_id":"n-6a03653d7a50bc7a","fingerprint":"f6-6bae7ffb6b8d3725","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6bae7ffb6b8d3725"},"other_id":"n-4113d906b9d95b00","other":{"id":"n-4113d906b9d95b00","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.","first_index":0,"fingerprint":"f6-6bae7ffb6b8d3725","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["121","198","81"],["1331","3267","2673","729"],["14641","47916","58806","32076","6561"]],"card_rows":[["121","198","81"],["1331","3267","2673","729"],["14641","47916","58806","32076","6561"],["161051","658845","1078110","882090","360855","59049"]],"row_sums_preview":["400","8000","160000","3200000","64000000","1280000000","25600000000","512000000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cfc5b48d32898a3e","steps":["reverse"],"edges":["e-9b28f01504767b0f"],"text":"A038301 \u2192 reverse"},"headline":"A038301 \u2192 reverse","chain":null}},{"id":"l-2996dd4fb5e3ef9a","kind":"fingerprint_related","node_a_id":"n-cfc5b48d32898a3e","node_b_id":"n-6a03653d7a50bc7a","fingerprint":"f6-6bae7ffb6b8d3725","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6bae7ffb6b8d3725"},"other_id":"n-cfc5b48d32898a3e","other":{"id":"n-cfc5b48d32898a3e","name":"Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","first_index":0,"fingerprint":"f6-6bae7ffb6b8d3725","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["81","198","121"],["729","2673","3267","1331"],["6561","32076","58806","47916","14641"]],"card_rows":[["81","198","121"],["729","2673","3267","1331"],["6561","32076","58806","47916","14641"],["59049","360855","882090","1078110","658845","161051"]],"row_sums_preview":["400","8000","160000","3200000","64000000","1280000000","25600000000","512000000000"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cfc5b48d32898a3e","steps":[],"edges":[],"text":"A038301"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","chain":null}},{"id":"l-4975ecbcc3de54a3","kind":"fingerprint_related","node_a_id":"n-3890397b83a674f9","node_b_id":"n-6a03653d7a50bc7a","fingerprint":"f6-6bae7ffb6b8d3725","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6bae7ffb6b8d3725"},"other_id":"n-3890397b83a674f9","other":{"id":"n-3890397b83a674f9","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.","first_index":0,"fingerprint":"f6-6bae7ffb6b8d3725","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["9"],["198","162"]],"card_rows":[["9"],["198","162"],["3267","5346","2187"],["47916","117612","96228","26244"]],"row_sums_preview":["0","9","360","10800","288000","7200000","172800000","4032000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4e096b5929fd3f2e","steps":["D"],"edges":["e-8b0042cde2dad276"],"text":"A038323 \u2192 D"},"headline":"A038323 \u2192 D","chain":null}},{"id":"l-4bd0912e6a8c3bc3","kind":"fingerprint_related","node_a_id":"n-b2ae2bce0e1523fc","node_b_id":"n-6a03653d7a50bc7a","fingerprint":"f6-6bae7ffb6b8d3725","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6bae7ffb6b8d3725"},"other_id":"n-b2ae2bce0e1523fc","other":{"id":"n-b2ae2bce0e1523fc","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.","first_index":0,"fingerprint":"f6-6bae7ffb6b8d3725","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["18","22"],["324","792","484"]],"card_rows":[["1"],["18","22"],["324","792","484"],["5832","21384","26136","10648"]],"row_sums_preview":["1","40","1600","64000","2560000","102400000","4096000000","163840000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b7cf68162d40a956","steps":["basis A013609"],"edges":["e-59f75239fe41171d"],"text":"A038277 \u2192 basis A013609"},"headline":"A038277 \u2192 basis A013609","chain":null}},{"id":"l-4d2fe51643177d9e","kind":"fingerprint_related","node_a_id":"n-ec8e2539152ade57","node_b_id":"n-6a03653d7a50bc7a","fingerprint":"f6-6bae7ffb6b8d3725","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6bae7ffb6b8d3725"},"other_id":"n-ec8e2539152ade57","other":{"id":"n-ec8e2539152ade57","name":"derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.","first_index":0,"fingerprint":"f6-6bae7ffb6b8d3725","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["198","162"],["3267","5346","2187"],["47916","117612","96228","26244"]],"card_rows":[["198","162"],["3267","5346","2187"],["47916","117612","96228","26244"],["658845","2156220","2646270","1443420","295245"]],"row_sums_preview":["360","10800","288000","7200000","172800000","4032000000","92160000000","2073600000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cfc5b48d32898a3e","steps":["reverse","D"],"edges":["e-9b28f01504767b0f","e-bf8ca1b977f10a19"],"text":"A038301 \u2192 reverse \u2192 D"},"headline":"A038301 \u2192 reverse \u2192 D","chain":null}},{"id":"l-5bc88b80ee06a07a","kind":"fingerprint_related","node_a_id":"n-1f17a7a2f7f54d5c","node_b_id":"n-6a03653d7a50bc7a","fingerprint":"f6-6bae7ffb6b8d3725","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6bae7ffb6b8d3725"},"other_id":"n-1f17a7a2f7f54d5c","other":{"id":"n-1f17a7a2f7f54d5c","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.","first_index":0,"fingerprint":"f6-6bae7ffb6b8d3725","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["22"],["792","968"]],"card_rows":[["22"],["792","968"],["21384","52272","31944"],["513216","1881792","2299968","937024"]],"row_sums_preview":["0","22","1760","105600","5632000","281600000","13516800000","630784000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b7cf68162d40a956","steps":["basis A013609","D"],"edges":["e-59f75239fe41171d","e-f8b03decdafd3042"],"text":"A038277 \u2192 basis A013609 \u2192 D"},"headline":"A038277 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-a9bf5a245132fc7c","kind":"fingerprint_related","node_a_id":"n-6a03653d7a50bc7a","node_b_id":"n-521827c5e15e2360","fingerprint":"f6-6bae7ffb6b8d3725","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6bae7ffb6b8d3725"},"other_id":"n-521827c5e15e2360","other":{"id":"n-521827c5e15e2360","name":"basis_map_A013609 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","first_index":0,"fingerprint":"f6-6bae7ffb6b8d3725","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["484","792","324"],["10648","26136","21384","5832"],["234256","766656","940896","513216","104976"]],"card_rows":[["484","792","324"],["10648","26136","21384","5832"],["234256","766656","940896","513216","104976"],["5153632","21083040","34499520","28226880","11547360","1889568"]],"row_sums_preview":["1600","64000","2560000","102400000","4096000000","163840000000","6553600000000","262144000000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4397e742863fdbe2","steps":["basis A013610","basis A013609"],"edges":["e-c80be26a429cc3fc","e-9512493cef2d9dab"],"text":"A038305 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A038305 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-aa37b61b78690ab1","kind":"fingerprint_related","node_a_id":"n-4e096b5929fd3f2e","node_b_id":"n-6a03653d7a50bc7a","fingerprint":"f6-6bae7ffb6b8d3725","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6bae7ffb6b8d3725"},"other_id":"n-4e096b5929fd3f2e","other":{"id":"n-4e096b5929fd3f2e","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.","first_index":0,"fingerprint":"f6-6bae7ffb6b8d3725","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["11","9"],["121","198","81"]],"card_rows":[["1"],["11","9"],["121","198","81"],["1331","3267","2673","729"]],"row_sums_preview":["1","20","400","8000","160000","3200000","64000000","1280000000"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A038323"],"derivation":{"root":"n-4e096b5929fd3f2e","steps":[],"edges":[],"text":"A038323"},"headline":"A038323: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.","chain":null}},{"id":"l-bc56fc78f7af2ffe","kind":"fingerprint_related","node_a_id":"n-95452b52aae7dbf2","node_b_id":"n-6a03653d7a50bc7a","fingerprint":"f6-6bae7ffb6b8d3725","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6bae7ffb6b8d3725"},"other_id":"n-95452b52aae7dbf2","other":{"id":"n-95452b52aae7dbf2","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","first_index":0,"fingerprint":"f6-6bae7ffb6b8d3725","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","11"],["81","198","121"]],"card_rows":[["1"],["9","11"],["81","198","121"],["729","2673","3267","1331"]],"row_sums_preview":["1","20","400","8000","160000","3200000","64000000","1280000000"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-366b2525683554ec","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-0da8e8e5bf86fd8d","e-bc0b86b7da83bbc9"],"text":"A038279 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A038279 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-c5c2eb795bb44953","kind":"fingerprint_related","node_a_id":"n-8156764a6e294c62","node_b_id":"n-6a03653d7a50bc7a","fingerprint":"f6-6bae7ffb6b8d3725","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6bae7ffb6b8d3725"},"other_id":"n-8156764a6e294c62","other":{"id":"n-8156764a6e294c62","name":"reversal of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.","first_index":0,"fingerprint":"f6-6bae7ffb6b8d3725","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["22","18"],["484","792","324"]],"card_rows":[["1"],["22","18"],["484","792","324"],["10648","26136","21384","5832"]],"row_sums_preview":["1","40","1600","64000","2560000","102400000","4096000000","163840000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b7cf68162d40a956","steps":["basis A013609","reverse"],"edges":["e-59f75239fe41171d","e-78c6f1d51c77ef97"],"text":"A038277 \u2192 basis A013609 \u2192 reverse"},"headline":"A038277 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-f1f8591610709cfe","kind":"fingerprint_related","node_a_id":"n-7153e756699bb014","node_b_id":"n-6a03653d7a50bc7a","fingerprint":"f6-6bae7ffb6b8d3725","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6bae7ffb6b8d3725"},"other_id":"n-7153e756699bb014","other":{"id":"n-7153e756699bb014","name":"reversal of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.","first_index":0,"fingerprint":"f6-6bae7ffb6b8d3725","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","11"],["81","198","121"]],"card_rows":[["1"],["9","11"],["81","198","121"],["729","2673","3267","1331"]],"row_sums_preview":["1","20","400","8000","160000","3200000","64000000","1280000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4e096b5929fd3f2e","steps":["reverse"],"edges":["e-381fdb511447f5c3"],"text":"A038323 \u2192 reverse"},"headline":"A038323 \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:          198,           242;\n  1:         2673,          6534,           3993;\n  2:        32076,        117612,         143748,          58564;\n  3:       360855,       1764180,        3234330,        2635380,          805255;\n  4:      3897234,      23816430,       58217940,       71155260,        43483770,        10629366;\n  5:     40920957,     300087018,      916932555,     1494260460,      1369738755,       669650058,       136410197;\n  6:    420901272,    3601044216,    13203828792,    26896688280,     32873730120,     24107402088,      9821534184,      1714871048;\n  7:   4261625379,   41669225928,   178251688692,   435726350136,    665693034930,    650899856376,    397772134452,    138904554888,    21221529219;\n  8:  42616253790,  468778791690,  2291807426040,  6535895252040,  11982474628740,  14645246768460,  11933164033560,   6250704969960,  1909937629710,   259374246010;\n  9: 421900912521, 5156566708590, 28361116897245, 92436232850280, 197710831374210, 289975886015508, 295345809830610, 206273264008680, 94541912670645, 25678050354990, 3138428376721;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = (9*n+18+(11*n+22)*x)*P(n-1,x) for n >= 1, with P(0,x) = 198 + 242*x.","mathematica":"P[0, x_] := 198 + 242*x;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n + 18 + (11*n + 22)*x)*P[n - 1, x])/(n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 198 + 242*x; for(n = 1, N, my(i = n + 1); v[i] = ((9*n + 18 + (11*n + 22)*x)*v[i - 1])/(n + 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-cfc5b48d32898a3e","steps":["D"],"edges":["e-4169abb442691e46"],"text":"A038301 \u2192 D"},"headline":"A038301 \u2192 D","chain":null}