{"id":"n-1cd850b4233af889","revision":0,"first_index":0,"fingerprint":"f6-2945f948d8e5ebc1","name":"theta_plus_one_squared of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^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":false,"rows":[["144","960","900"],["1728","17280","32400","16000"],["20736","276480","777600","768000","250000"],["248832","4147200","15552000","23040000","15000000","3600000"],["2985984","59719680","279936000","552960000","540000000","259200000","49000000"],["35831808","836075520","4702924800","11612160000","15120000000","10886400000","4116000000","640000000"],["429981696","11466178560","75246796800","222953472000","362880000000","348364800000","197568000000","61440000000","8100000000"],["5159780352","154793410560","1160950579200","4013162496000","7838208000000","9405849600000","7112448000000","3317760000000","874800000000","100000000000"],["61917364224","2063912140800","17414258688000","68797071360000","156764160000000","225740390400000","213373440000000","132710400000000","52488000000000","12000000000000","1210000000000"],["743008370688","27243640258560","255409127424000","1135151677440000","2956124160000000","4966288588800000","5633058816000000","4379443200000000","2309472000000000","792000000000000","159720000000000","14400000000000"],["8916100448256","356644017930240","3677891434905600","18162426839040000","53210234880000000","102163650969600000","135193411584000000","126127964160000000","83140992000000000","38016000000000000","11499840000000000","2073600000000000","169000000000000"],["106993205379072","4636372233093120","52159187622297600","283333858689024000","922310737920000000","1992191193907200000","3012881743872000000","3279327068160000000","2593998950400000000","1482624000000000000","597991680000000000","161740800000000000","26364000000000000","1960000000000000"],["1283918464548864","59916195012280320","730228626712166400","4327280750886912000","15494820397056000000","37187568952934400000","63270516621312000000","78703849635840000000","72631970611200000000","49816166400000000000","25115650560000000000","9057484800000000000","2214576000000000000","329280000000000000","22500000000000000"],["15407021574586368","770351078729318400","10110857908322304000","64909211263303680000","253551606497280000000","669376241152819200000","1265410332426240000000","1770836616806400000000","1867679244288000000000","1494484992000000000000","904163420160000000000","407586816000000000000","132874560000000000000","29635200000000000000","4050000000000000000","256000000000000000"],["184884258895036416","9860493807735275520","138663194171277312000","958659120196485120000","4056825703956480000000","11683658027394662400000","24295878382583808000000","37777847825203200000000","44824301862912000000000","40991588352000000000000","28933229445120000000000","15651333734400000000000","6377978880000000000000","1896652800000000000000","388800000000000000000","49152000000000000000","2890000000000000000"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["2004","67408","2092816","61588032","1743801664","47949392128","1288449229056","33983131866112","882623549953024","22629054218293248","573800571985924096","14411626586008993792","358963093469418307584","8875710856602669088768","218037504706004543143936"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["144","960","900"]],"recurrence":{"denominator":{"coeffs":[["36","150","100"],["0","90","100"],["0","0","25"]]},"terms":[{"coeff":{"coeffs":[["432","3240","5100","2250"],["0","1080","2700","1500"],["0","0","300","250"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193756,"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-65eb9b8e41b1be84"]},{"id":193757,"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-65eb9b8e41b1be84"]},{"id":193758,"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-65eb9b8e41b1be84"]},{"id":193759,"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-65eb9b8e41b1be84"]}],"region_exact":{},"incoming_edges":[{"id":"e-65eb9b8e41b1be84","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-1cd850b4233af889","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"5ec145d5d49ea367da4acfec7d04a1f62d4443b45c887d050b7e800b53b69366","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-ec89fb3fc20684fc","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^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-34c4172a9a85b14a","kind":"fingerprint_related","node_a_id":"n-599f20cbf25e8980","node_b_id":"n-1cd850b4233af889","fingerprint":"f6-2945f948d8e5ebc1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2945f948d8e5ebc1"},"other_id":"n-599f20cbf25e8980","other":{"id":"n-599f20cbf25e8980","name":"theta_plus_one_squared of basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-2945f948d8e5ebc1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["36","240","225"],["216","2160","4050","2000"],["1296","17280","48600","48000","15625"]],"card_rows":[["36","240","225"],["216","2160","4050","2000"],["1296","17280","48600","48000","15625"],["7776","129600","486000","720000","468750","112500"]],"row_sums_preview":["501","8426","130801","1924626","27246901","374604626","5033004801","66373304426"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-7ae18e4d322020fd","e-f80ee1ef72ff7501"],"text":"A013612 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A013612 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-5c212eb0362283ae","kind":"fingerprint_related","node_a_id":"n-9922e8d8ef3f8572","node_b_id":"n-1cd850b4233af889","fingerprint":"f6-2945f948d8e5ebc1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2945f948d8e5ebc1"},"other_id":"n-9922e8d8ef3f8572","other":{"id":"n-9922e8d8ef3f8572","name":"theta_plus_one_squared of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-2945f948d8e5ebc1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["12","40"],["144","960","900"]],"card_rows":[["1"],["12","40"],["144","960","900"],["1728","17280","32400","16000"]],"row_sums_preview":["1","52","2004","67408","2092816","61588032","1743801664","47949392128"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-e169cb33ba7b2c8c","e-5095495b025f8927"],"text":"A038271 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038271 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-a256bbcb89f07b46","kind":"fingerprint_related","node_a_id":"n-f8c750a328bdc1fd","node_b_id":"n-1cd850b4233af889","fingerprint":"f6-2945f948d8e5ebc1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2945f948d8e5ebc1"},"other_id":"n-f8c750a328bdc1fd","other":{"id":"n-f8c750a328bdc1fd","name":"theta_plus_one_squared of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.","first_index":0,"fingerprint":"f6-2945f948d8e5ebc1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","20"],["36","240","225"]],"card_rows":[["1"],["6","20"],["36","240","225"],["216","2160","4050","2000"]],"row_sums_preview":["1","26","501","8426","130801","1924626","27246901","374604626"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e6ebe74518d42cbe","steps":["(\u03b8+1)\u00b2"],"edges":["e-2f78870c2090199f"],"text":"A038259 \u2192 (\u03b8+1)\u00b2"},"headline":"A038259 \u2192 (\u03b8+1)\u00b2","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:          144,            960,             900;\n  1:         1728,          17280,           32400,            16000;\n  2:        20736,         276480,          777600,           768000,           250000;\n  3:       248832,        4147200,        15552000,         23040000,         15000000,          3600000;\n  4:      2985984,       59719680,       279936000,        552960000,        540000000,        259200000,         49000000;\n  5:     35831808,      836075520,      4702924800,      11612160000,      15120000000,      10886400000,       4116000000,        640000000;\n  6:    429981696,    11466178560,     75246796800,     222953472000,     362880000000,     348364800000,     197568000000,      61440000000,       8100000000;\n  7:   5159780352,   154793410560,   1160950579200,    4013162496000,    7838208000000,    9405849600000,    7112448000000,    3317760000000,     874800000000,    100000000000;\n  8:  61917364224,  2063912140800,  17414258688000,   68797071360000,  156764160000000,  225740390400000,  213373440000000,  132710400000000,   52488000000000,  12000000000000,   1210000000000;\n  9: 743008370688, 27243640258560, 255409127424000, 1135151677440000, 2956124160000000, 4966288588800000, 5633058816000000, 4379443200000000, 2309472000000000, 792000000000000, 159720000000000, 14400000000000;\n","recurrence_sentence":"The row polynomials satisfy (36+(90*n+150)*x+(25*n^2+100*n+100)*x^2)*P(n,x) = (432+(1080*n+3240)*x+(300*n^2+2700*n+5100)*x^2+(250*n^2+1500*n+2250)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 960*x + 900*x^2.","mathematica":"P[0, x_] := 144 + 960*x + 900*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((432 + (1080*n + 3240)*x + (300*n^2 + 2700*n + 5100)*x^2 + (250*n^2 + 1500*n + 2250)*x^3)*P[n - 1, x])/(36 + (90*n + 150)*x + (25*n^2 + 100*n + 100)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 144 + 960*x + 900*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((432 + (1080*n + 3240)*x + (300*n^2 + 2700*n + 5100)*x^2 + (250*n^2 + 1500*n + 2250)*x^3)*v[i - 1])/(36 + (90*n + 150)*x + (25*n^2 + 100*n + 100)*x^2)); 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-ec89fb3fc20684fc","steps":["(\u03b8+1)\u00b2"],"edges":["e-65eb9b8e41b1be84"],"text":"A038336 \u2192 (\u03b8+1)\u00b2"},"headline":"A038336 \u2192 (\u03b8+1)\u00b2","chain":null}
