{"id":"n-3890397b83a674f9","revision":0,"first_index":0,"fingerprint":"f6-6bae7ffb6b8d3725","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^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":[[],["9"],["198","162"],["3267","5346","2187"],["47916","117612","96228","26244"],["658845","2156220","2646270","1443420","295245"],["8696754","35577630","58217940","47632860","19486170","3188646"],["111608343","547895502","1120695345","1222576740","750217545","245525742","33480783"],["1403076312","8035800696","19724238072","26896688280","22006381320","10803132648","2946308904","344373768"],["17363069361","113649181272","325449928188","532554427944","544657937670","356503377384","145842290748","34093003032","3486784401"],["212215292190","1562676242490","5114213157240","9763497845640","11982474628740","9803842878060","5347550660760","1875115166760","383546284110","34867844010"],["2567805035499","21009313926810","77352474003255","168769034188920","241646571679590","237252997649052","161763407487990","75629645059320","23204550188655","4219009125210","345191655699"],["30813660425988","277322943833892","1134502952047740","2784689064117180","4556763923100840","5219565948279144","4270553957682936","2495778286957560","1021000208300820","278454602263860","45565298552268","3389154437772"],["367196120076357","3605198269840596","16223392214282682","44245615129861860","81452155125427515","106628275800559656","101781535991443308","71379259006986216","36500757446754315","13273002707910660","3257918846487162","484649084601396","33044255768277"],["4349861730135306","46266711129620982","227127490999957548","681382472999872644","1393736876590648590","2052594309160773378","2239193791811752776","1832067647845979544","1124223329360032902","511010604254560410","167239834119674316","37317979514307492","5088815388314658","320275094369454"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","9","360","10800","288000","7200000","172800000","4032000000","92160000000","2073600000000","46080000000000","1013760000000000","22118400000000000","479232000000000000","10321920000000000000"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["9"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["11","9"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190403,"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-8b0042cde2dad276"]},{"id":190404,"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-8b0042cde2dad276"]},{"id":190405,"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-8b0042cde2dad276"]},{"id":190406,"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-8b0042cde2dad276"]}],"region_exact":{},"incoming_edges":[{"id":"e-8b0042cde2dad276","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-3890397b83a674f9","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":"fbe74813c7018b474f0451ed3870f1890765ff37fd5cb2bb943079e0919d0d3e","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-4e096b5929fd3f2e","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^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-28c8f1ce98c5ffe1","kind":"fingerprint_related","node_a_id":"n-3890397b83a674f9","node_b_id":"n-1f17a7a2f7f54d5c","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-30f56335ec929f9c","kind":"fingerprint_related","node_a_id":"n-4e096b5929fd3f2e","node_b_id":"n-3890397b83a674f9","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-3b8da214ac9cc1ab","kind":"fingerprint_related","node_a_id":"n-cfc5b48d32898a3e","node_b_id":"n-3890397b83a674f9","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-402e904d66cb195e","kind":"fingerprint_related","node_a_id":"n-b2ae2bce0e1523fc","node_b_id":"n-3890397b83a674f9","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-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-6a03653d7a50bc7a","other":{"id":"n-6a03653d7a50bc7a","name":"derivative of 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":[["198","242"],["2673","6534","3993"],["32076","117612","143748","58564"]],"card_rows":[["198","242"],["2673","6534","3993"],["32076","117612","143748","58564"],["360855","1764180","3234330","2635380","805255"]],"row_sums_preview":["440","13200","352000","8800000","211200000","4928000000","112640000000","2534400000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cfc5b48d32898a3e","steps":["D"],"edges":["e-4169abb442691e46"],"text":"A038301 \u2192 D"},"headline":"A038301 \u2192 D","chain":null}},{"id":"l-6437fbc6843ee783","kind":"fingerprint_related","node_a_id":"n-3890397b83a674f9","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-903380095b85f976","kind":"fingerprint_related","node_a_id":"n-4113d906b9d95b00","node_b_id":"n-3890397b83a674f9","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-b34e9f999376c4be","kind":"fingerprint_related","node_a_id":"n-3890397b83a674f9","node_b_id":"n-8156764a6e294c62","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-b3d170845cfa8cad","kind":"fingerprint_related","node_a_id":"n-3890397b83a674f9","node_b_id":"n-ec8e2539152ade57","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-c1fb46d83f4e884f","kind":"fingerprint_related","node_a_id":"n-95452b52aae7dbf2","node_b_id":"n-3890397b83a674f9","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-c7140e54c47d6aed","kind":"fingerprint_related","node_a_id":"n-3890397b83a674f9","node_b_id":"n-7153e756699bb014","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:           0;\n  1:           9;\n  2:         198,          162;\n  3:        3267,         5346,         2187;\n  4:       47916,       117612,        96228,        26244;\n  5:      658845,      2156220,      2646270,      1443420,       295245;\n  6:     8696754,     35577630,     58217940,     47632860,     19486170,      3188646;\n  7:   111608343,    547895502,   1120695345,   1222576740,    750217545,    245525742,     33480783;\n  8:  1403076312,   8035800696,  19724238072,  26896688280,  22006381320,  10803132648,   2946308904,   344373768;\n  9: 17363069361, 113649181272, 325449928188, 532554427944, 544657937670, 356503377384, 145842290748, 34093003032, 3486784401;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (11*n+9*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 9.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 9;\nP[n_, x_] := P[n, x] = Expand[Together[((11*n + 9*n*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] = 0; v[2] = 9; for(n = 2, N, my(i = n + 1); v[i] = ((11*n + 9*n*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-4e096b5929fd3f2e","steps":["D"],"edges":["e-8b0042cde2dad276"],"text":"A038323 \u2192 D"},"headline":"A038323 \u2192 D","chain":null}