{"id":"n-eea50a67a563377c","revision":0,"first_index":0,"fingerprint":"f6-efa2c9a7f1c36a29","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(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":[["1012","968"],["34914","66792","31944"],["1070696","3072432","2938848","937024"],["30782510","117776560","168983760","107757760","25768160"],["849597276","4063291320","7773252960","7435285440","3556006080","680279424"],["22797526906","130837980504","312873431640","399026985280","286258489440","109524987264","17460505216"],["599249278672","4012364735456","11513742284352","18355241322880","17557187352320","10076298828288","3212732959744","439006988288"],["15505575085638","118651357177056","397224108810144","759906990767232","908584445482560","695264619151872","332517861333504","90874446575616","10865422960128"],["396253585521860","3411226518840360","13051649289476160","29129767979410560","41794884492197760","39977715601232640","25493036035568640","10450561356195840","2499047280829440","265599227914240"],["10025215713703058","95893367696290120","412758408779683560","1052833042684410240","1762350962754338880","2022872409422371584","1612434529249716480","881330674372515840","316129481024924160","67196604662302720","6427501315524608"],["251541776089276728","2646656948417607312","12657924535910295840","36322739972612153280","69486980817171075840","93052130833429092864","89006386014584349696","60811816531703592960","29083912254293022720","9273131443397775360","1773990363084791808","154260031572590592"],["6267582587557811806","71940947961533144208","378471943623717845616","1206722139090114870080","2597075908041768959520","3974655302742185538048","4435484903060120093184","3636546628595874859008","2174022441008403448320","924222100525311610880","265211559281176375296","46123749440204587008","3676530752480075776"],["155243199476431953964","1930415436967806036248","11078905986076104208032","38856452878701698816576","92917604709938844996160","159979875935372967906432","204032305540765524286464","195161335734645284100096","140007045200941182071808","74399879092287584675840","28466040696179597615104","7425923659872938508288","1183842902298584399872","87105497827989487616"],["3825635987097787436970","51230255827222544808120","318518547099687995980920","1218679658468371462883520","3205657362492890152367520","6132561910855963769746560","8798893176445513234853760","9618665832636089002076160","8050405099054117969128960","5133591657367843342632960","2455196010045490294302720","853981220885387928453120","204212900646505808977920","30051396750656373227520","2053201020231180779520"],["93855602883465718453664","1346623867458421177813440","9016525025591167886229120","37372842859696724861761280","107243809945216688733749760","225678278319499466726673408","359776965436883207825131520","442458628301260094095503360","423221296635987916091351040","314860288318561058348154880","180702426339348085660680192","78566272321455689417687040","25050115812638045901291520","5529457002120772673863680","755577975445074526863360","48181783941425042292736"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["1980","133650","8019000","451068750","24357712500","1278779906250","65765823750000","3329394827343750","166469741367187500","8240252197675781250","404521471522265625000","19720421736710449218750","955681976471352539062500","46077523865583068847656250","2211721145547987304687500000"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1012","968"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["46","44"],["23","22"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194962,"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-fb2442c67dee93f1"]},{"id":194963,"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-fb2442c67dee93f1"]},{"id":194964,"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-fb2442c67dee93f1"]},{"id":194965,"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-fb2442c67dee93f1"]}],"region_exact":{},"incoming_edges":[{"id":"e-fb2442c67dee93f1","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-eea50a67a563377c","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":"e96f06faddb6f5f905df4a6605bfe05f7b551f905846f5d484ca1ff13cabd77d","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-71831045520c8a2d","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(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-fa3ebac699a39308","kind":"fingerprint_related","node_a_id":"n-71831045520c8a2d","node_b_id":"n-eea50a67a563377c","fingerprint":"f6-efa2c9a7f1c36a29","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-efa2c9a7f1c36a29"},"other_id":"n-71831045520c8a2d","other":{"id":"n-71831045520c8a2d","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^j.","first_index":0,"fingerprint":"f6-efa2c9a7f1c36a29","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["529","1012","484"],["12167","34914","33396","10648"],["279841","1070696","1536216","979616","234256"]],"card_rows":[["529","1012","484"],["12167","34914","33396","10648"],["279841","1070696","1536216","979616","234256"],["6436343","30782510","58888280","56327920","26939440","5153632"]],"row_sums_preview":["2025","91125","4100625","184528125","8303765625","373669453125","16815125390625","756680642578125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-344e4bfa9933088a","steps":["basis A013609"],"edges":["e-9947ebaa7f991114"],"text":"A038337 \u2192 basis A013609"},"headline":"A038337 \u2192 basis A013609","chain":null}},{"id":"l-feb6ebe8525e6d23","kind":"fingerprint_related","node_a_id":"n-eea50a67a563377c","node_b_id":"n-6f980c37a6fafca3","fingerprint":"f6-efa2c9a7f1c36a29","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-efa2c9a7f1c36a29"},"other_id":"n-6f980c37a6fafca3","other":{"id":"n-6f980c37a6fafca3","name":"reversal of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^j.","first_index":0,"fingerprint":"f6-efa2c9a7f1c36a29","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["484","1012","529"],["10648","33396","34914","12167"],["234256","979616","1536216","1070696","279841"]],"card_rows":[["484","1012","529"],["10648","33396","34914","12167"],["234256","979616","1536216","1070696","279841"],["5153632","26939440","56327920","58888280","30782510","6436343"]],"row_sums_preview":["2025","91125","4100625","184528125","8303765625","373669453125","16815125390625","756680642578125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-344e4bfa9933088a","steps":["basis A013609","reverse"],"edges":["e-9947ebaa7f991114","e-ea75b770002f08bd"],"text":"A038337 \u2192 basis A013609 \u2192 reverse"},"headline":"A038337 \u2192 basis A013609 \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:              1012,               968;\n  1:             34914,             66792,              31944;\n  2:           1070696,           3072432,            2938848,              937024;\n  3:          30782510,         117776560,          168983760,           107757760,            25768160;\n  4:         849597276,        4063291320,         7773252960,          7435285440,          3556006080,           680279424;\n  5:       22797526906,      130837980504,       312873431640,        399026985280,        286258489440,        109524987264,         17460505216;\n  6:      599249278672,     4012364735456,     11513742284352,      18355241322880,      17557187352320,      10076298828288,       3212732959744,       439006988288;\n  7:    15505575085638,   118651357177056,    397224108810144,     759906990767232,     908584445482560,     695264619151872,     332517861333504,     90874446575616,     10865422960128;\n  8:   396253585521860,  3411226518840360,  13051649289476160,   29129767979410560,   41794884492197760,   39977715601232640,   25493036035568640,  10450561356195840,   2499047280829440,   265599227914240;\n  9: 10025215713703058, 95893367696290120, 412758408779683560, 1052833042684410240, 1762350962754338880, 2022872409422371584, 1612434529249716480, 881330674372515840, 316129481024924160, 67196604662302720, 6427501315524608;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = (23*n+46+(22*n+44)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1012 + 968*x.","mathematica":"P[0, x_] := 1012 + 968*x;\nP[n_, x_] := P[n, x] = Expand[Together[((23*n + 46 + (22*n + 44)*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] = 1012 + 968*x; for(n = 1, N, my(i = n + 1); v[i] = ((23*n + 46 + (22*n + 44)*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-344e4bfa9933088a","steps":["basis A013609","D"],"edges":["e-9947ebaa7f991114","e-fb2442c67dee93f1"],"text":"A038337 \u2192 basis A013609 \u2192 D"},"headline":"A038337 \u2192 basis A013609 \u2192 D","chain":null}