{"id":"n-4e0745b0dd22f6a8","revision":0,"first_index":0,"fingerprint":"f6-11954419b7b7488a","name":"theta_plus_one_squared of basis_map_A013610 of Unsigned 2-Stirling numbers of the first kind.","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":[["12","84","81"],["60","564","972","432"],["360","4104","9639","7776","2025"],["2520","33048","95175","105840","50625","8748"],["20160","294624","984474","1354320","901125","288684","35721"],["181440","2893536","10848978","17439408","14458500","6491016","1500282","139968"],["1814400","31112640","128021148","232255296","226332225","127370880","41507802","7278336","531441"],["19958400","364011840","1618242948","3237587712","3578351175","2378834892","976683582","242704512","33480783","1968300"],["239500800","4607642880","21875995296","47481681600","58116406500","44004495780","21433778793","6739459200","1323288090","147622500","7144929"],["3113510400","62773367040","315489528288","733933835712","978083667000","823121321220","458324148744","171598388544","42791629320","6820159500","628753752","25509168"],["43589145600","916189263360","4840573623552","11957684517504","17133486192900","15749019938520","9777616810731","4198259818368","1250620441983","253969749000","33559731513","2601935136","89813529"],["653837184000","14265908697600","78792881880960","205181660421504","313053939069300","310251959431128","210972750243255","101286191717568","34699574377611","8441473797900","1425306161565","158845589136","10508182893","312487308"],["10461394944000","236100585369600","1356980993804160","3703135270109184","5970652058334600","6316424367677424","4642426171569186","2447247803128128","939764199219417","263580522905700","53447448471417","7630225804656","727399771371","41560811964","1076168025"],["177843714048000","4139246690611200","24662355845915520","70190531558811648","118859531570325000","133172431142521680","104713311084692310","59793943774428864","25267885389232200","7961477034654000","1865403922161780","320534481155616","39230549467200","3237368510880","161425203750","3673320192"],["3201186852864000","76640564999577600","471862320388104960","1394962132570159104","2468489684947779600","2910576936949194240","2428627026689758440","1486596206883820032","681852067274714193","236891347324632000","62477432234706060","12429575159758848","1834698376144998","194767089330240","14054754406500","617117792256","12440502369"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["177","2028","23904","295956","3879108","53953128","796224168","12451824144","205977016368","3596703818688","66157260981696","1279195149241728","25948621078088832","551167129225059840","12236650632053245440"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["12","84","81"],["60","564","972","432"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-9","-36","81"],["-6","-12","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","36","36"],["0","0","12","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["7","12"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-12"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189571,"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-7dcedfa4fdcff1c2"]},{"id":189572,"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-7dcedfa4fdcff1c2"]},{"id":189573,"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-7dcedfa4fdcff1c2"]},{"id":189574,"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-7dcedfa4fdcff1c2"]}],"region_exact":{},"incoming_edges":[{"id":"e-7dcedfa4fdcff1c2","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-4e0745b0dd22f6a8","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":"fd01852f43b0560ad66a2f1df2e945b6ab7357aa0c77a0b42ddb16a62b83ad28","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-0d23d1c7119dd8ef","role":"sequence","name":"basis_map_A013610 of Unsigned 2-Stirling numbers of the first kind.","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-582645e767b67c6a","kind":"fingerprint_related","node_a_id":"n-e3fc2db31bb04e25","node_b_id":"n-4e0745b0dd22f6a8","fingerprint":"f6-11954419b7b7488a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11954419b7b7488a"},"other_id":"n-e3fc2db31bb04e25","other":{"id":"n-e3fc2db31bb04e25","name":"theta_plus_one_squared of basis_map_A013610 of Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).","first_index":0,"fingerprint":"f6-11954419b7b7488a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","12"],["12","84","81"]],"card_rows":[["1"],["3","12"],["12","84","81"],["60","564","972","432"]],"row_sums_preview":["1","15","177","2028","23904","295956","3879108","53953128"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-914700b1060feeb2","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-8500cb477569290a","e-6720947fc06a7902"],"text":"A136124 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A136124 \u2192 basis A013610 \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:         12,          84,           81;\n  1:         60,         564,          972,          432;\n  2:        360,        4104,         9639,         7776,         2025;\n  3:       2520,       33048,        95175,       105840,        50625,         8748;\n  4:      20160,      294624,       984474,      1354320,       901125,       288684,        35721;\n  5:     181440,     2893536,     10848978,     17439408,     14458500,      6491016,      1500282,       139968;\n  6:    1814400,    31112640,    128021148,    232255296,    226332225,    127370880,     41507802,      7278336,      531441;\n  7:   19958400,   364011840,   1618242948,   3237587712,   3578351175,   2378834892,    976683582,    242704512,    33480783,    1968300;\n  8:  239500800,  4607642880,  21875995296,  47481681600,  58116406500,  44004495780,  21433778793,   6739459200,  1323288090,  147622500,   7144929;\n  9: 3113510400, 62773367040, 315489528288, 733933835712, 978083667000, 823121321220, 458324148744, 171598388544, 42791629320, 6820159500, 628753752, 25509168;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+6*n+9+(12*n+36)*x-81*x^2)*P(n-2,x) + ((12*n+36)*x^2+36*x^3)*d/dx P(n-2,x) + (2*n+7+12*x)*P(n-1,x) - 12*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 12 + 84*x + 81*x^2, P(1,x) = 60 + 564*x + 972*x^2 + 432*x^3.","mathematica":"P[0, x_] := 12 + 84*x + 81*x^2;\nP[1, x_] := 60 + 564*x + 972*x^2 + 432*x^3;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n + 9 + (12*n + 36)*x - 81*x^2)*P[n - 2, x] + ((12*n + 36)*x^2 + 36*x^3)*D[P[n - 2, x], x] + (2*n + 7 + 12*x)*P[n - 1, x] - 12*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 12 + 84*x + 81*x^2; v[2] = 60 + 564*x + 972*x^2 + 432*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n + 9 + (12*n + 36)*x - 81*x^2)*v[i - 2] + ((12*n + 36)*x^2 + 36*x^3)*deriv(v[i - 2], x) + (2*n + 7 + 12*x)*v[i - 1] - 12*x^2*deriv(v[i - 1], x)); 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-c237629cf5fdbb07","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-36015c56c1237184","e-7dcedfa4fdcff1c2"],"text":"A143491 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A143491 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}
