{"id":"n-56f92c5c3b659db9","revision":0,"first_index":0,"fingerprint":"f6-b6561930bb158c66","name":"derivative of Triangle of coefficients in expansion of (1+11x)^n.","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":[["22","242"],["33","726","3993"],["44","1452","15972","58564"],["55","2420","39930","292820","805255"],["66","3630","79860","878460","4831530","10629366"],["77","5082","139755","2049740","16910355","74405562","136410197"],["88","6776","223608","4099480","45094280","297622248","1091281576","1714871048"],["99","8712","335412","7379064","101462130","892866744","4910767092","15433839432","21221529219"],["110","10890","479160","12298440","202924260","2232166860","16369223640","77169197160","212215292190","259374246010"],["121","13310","658845","19326120","372027810","4910767092","45015365010","282953722920","1167184107045","2853116706110","3138428376721"],["132","15972","878460","28989180","637761960","9821534184","108036876024","848861168760","4668736428180","17118700236660","37661140520652","37661140520652"],["143","18876","1141998","41873260","1036363185","18239992056","234079898052","2207039038776","15173393391585","74181034358860","244797413384238","489594826768476","448795257871103"],["154","22022","1453452","58622564","1612120510","31919986098","468159796104","5149757757144","42485501496438","259633620256010","1142387929126444","3427163787379332","6283133610195442","5316497670165374"],["165","25410","1816815","79939860","2418180765","53199976830","877799617695","11035195193880","106213753741095","778900860768030","4283954734224165","17135818936896660","47123502076465815","79747465052480610","62658722541234765"],["176","29040","2236080","106586480","3517353840","85119962928","1560532653680","22070390387760","242774294265360","2077068962048080","13708655149517328","68543275747586640","251325344407817680","637979720419844880","1002539560659756240","735195677817154576"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["264","4752","76032","1140480","16422912","229920768","3153199104","42568187904","567575838720","7492001071104","98077104930816","1275002364100608","16476953628377088","211846546650562560","2711635797127200768"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["22","242"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","22"],["1","11"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191475,"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-25d575244d8ba049"]},{"id":191476,"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-25d575244d8ba049"]},{"id":191477,"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-25d575244d8ba049"]},{"id":191478,"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-25d575244d8ba049"]}],"region_exact":{},"incoming_edges":[{"id":"e-25d575244d8ba049","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-56f92c5c3b659db9","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":"a454f8bf0cc18df6580bce9dab48af55d79e83e1bf4f4e1d5eee33a2037af7b9","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-b1e393c17b0898c2","role":"sequence","name":"Triangle of coefficients in expansion of (1+11x)^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-4238a3d67e3f1124","kind":"fingerprint_related","node_a_id":"n-56f92c5c3b659db9","node_b_id":"n-6f60b9bdcd89fe3a","fingerprint":"f6-b6561930bb158c66","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b6561930bb158c66"},"other_id":"n-6f60b9bdcd89fe3a","other":{"id":"n-6f60b9bdcd89fe3a","name":"reversal of Triangle of coefficients in expansion of (1+11x)^n.","first_index":0,"fingerprint":"f6-b6561930bb158c66","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["121","22","1"],["1331","363","33","1"],["14641","5324","726","44","1"]],"card_rows":[["121","22","1"],["1331","363","33","1"],["14641","5324","726","44","1"],["161051","73205","13310","1210","55","1"]],"row_sums_preview":["144","1728","20736","248832","2985984","35831808","429981696","5159780352"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b1e393c17b0898c2","steps":["reverse"],"edges":["e-78acc0aaea72e8ff"],"text":"A013618 \u2192 reverse"},"headline":"A013618 \u2192 reverse","chain":null}},{"id":"l-a845136ad3835c1e","kind":"fingerprint_related","node_a_id":"n-b1e393c17b0898c2","node_b_id":"n-56f92c5c3b659db9","fingerprint":"f6-b6561930bb158c66","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b6561930bb158c66"},"other_id":"n-b1e393c17b0898c2","other":{"id":"n-b1e393c17b0898c2","name":"Triangle of coefficients in expansion of (1+11x)^n.","first_index":0,"fingerprint":"f6-b6561930bb158c66","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","22","121"],["1","33","363","1331"],["1","44","726","5324","14641"]],"card_rows":[["1","22","121"],["1","33","363","1331"],["1","44","726","5324","14641"],["1","55","1210","13310","73205","161051"]],"row_sums_preview":["144","1728","20736","248832","2985984","35831808","429981696","5159780352"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b1e393c17b0898c2","steps":[],"edges":[],"text":"A013618"},"headline":"Triangle of coefficients in expansion of (1+11x)^n.","chain":null}},{"id":"l-b87fe50ec1d6a901","kind":"fingerprint_related","node_a_id":"n-5914456fbb478c2c","node_b_id":"n-56f92c5c3b659db9","fingerprint":"f6-b6561930bb158c66","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b6561930bb158c66"},"other_id":"n-5914456fbb478c2c","other":{"id":"n-5914456fbb478c2c","name":"reversal of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","first_index":0,"fingerprint":"f6-b6561930bb158c66","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","11"],["1","22","121"]],"card_rows":[["1"],["1","11"],["1","22","121"],["1","33","363","1331"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["basis A007318","reverse"],"edges":["e-34d290d6207a0155","e-2f0a45ab280d0042"],"text":"A038303 \u2192 basis A007318 \u2192 reverse"},"headline":"A038303 \u2192 basis A007318 \u2192 reverse","chain":null}},{"id":"l-d3178ae2368f38f5","kind":"fingerprint_related","node_a_id":"n-31f0871ac8eccdd6","node_b_id":"n-56f92c5c3b659db9","fingerprint":"f6-b6561930bb158c66","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b6561930bb158c66"},"other_id":"n-31f0871ac8eccdd6","other":{"id":"n-31f0871ac8eccdd6","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","first_index":0,"fingerprint":"f6-b6561930bb158c66","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["11","1"],["121","22","1"]],"card_rows":[["1"],["11","1"],["121","22","1"],["1331","363","33","1"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["basis A007318"],"edges":["e-34d290d6207a0155"],"text":"A038303 \u2192 basis A007318"},"headline":"A038303 \u2192 basis A007318","chain":null}},{"id":"l-de2685a23234d750","kind":"fingerprint_related","node_a_id":"n-5fa66095534ffc89","node_b_id":"n-56f92c5c3b659db9","fingerprint":"f6-b6561930bb158c66","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b6561930bb158c66"},"other_id":"n-5fa66095534ffc89","other":{"id":"n-5fa66095534ffc89","name":"derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","first_index":0,"fingerprint":"f6-b6561930bb158c66","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["22","2"]],"card_rows":[["1"],["22","2"],["363","66","3"],["5324","1452","132","4"]],"row_sums_preview":["0","1","24","432","6912","103680","1492992","20901888"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["basis A007318","D"],"edges":["e-34d290d6207a0155","e-72c0eb18de28d615"],"text":"A038303 \u2192 basis A007318 \u2192 D"},"headline":"A038303 \u2192 basis A007318 \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:  22,   242;\n  1:  33,   726,   3993;\n  2:  44,  1452,  15972,    58564;\n  3:  55,  2420,  39930,   292820,    805255;\n  4:  66,  3630,  79860,   878460,   4831530,   10629366;\n  5:  77,  5082, 139755,  2049740,  16910355,   74405562,   136410197;\n  6:  88,  6776, 223608,  4099480,  45094280,  297622248,  1091281576,   1714871048;\n  7:  99,  8712, 335412,  7379064, 101462130,  892866744,  4910767092,  15433839432,   21221529219;\n  8: 110, 10890, 479160, 12298440, 202924260, 2232166860, 16369223640,  77169197160,  212215292190,  259374246010;\n  9: 121, 13310, 658845, 19326120, 372027810, 4910767092, 45015365010, 282953722920, 1167184107045, 2853116706110, 3138428376721;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = (n+2+(11*n+22)*x)*P(n-1,x) for n >= 1, with P(0,x) = 22 + 242*x.","mathematica":"P[0, x_] := 22 + 242*x;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 2 + (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] = 22 + 242*x; for(n = 1, N, my(i = n + 1); v[i] = ((n + 2 + (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-b1e393c17b0898c2","steps":["D"],"edges":["e-25d575244d8ba049"],"text":"A013618 \u2192 D"},"headline":"A013618 \u2192 D","chain":null}